Evaluation of computational methods of paleostress analysis using fault-striation data

E V A L U A T IO N O F C O M P U T A T IO N A L M E T H O D S O F

P A L E O S T R E S S A N A L Y S IS U S IN G F A U L T -S T R IA T IO N D A T A

A b s t ra c t o f

a t h e s i s p r e s e n t e d t o t h e F a c u l t y

o f th e S t a t e U n i v e r s i ty o f N e w Y o r k

a t A lb a n y

i n p a r t ia l fu l f i l l m e n t o f t h e r e q u i r e m e n t s

fo r th e d e g re e o f

M a s te r o f S c ie n c e

C o l l e g e o f S c ie n c e a n d M a th e m a t i c s

D e p a r tm e n t o f G e o lo g ic a l S c ie n c e s

S t e v e n H e n ry S c h im m r ic h

1 9 9 1

A B S T R A C T

O v e r t h e p a s t 1 2 y e a r s , m a n y d i f f e r e n t c o m p u ta t io n a l m e th o d s o r v a r i a t i o n s o f e x i s t i n g

m e th o d s h a v e b e e n p ro p o s e d fo r d e t e rm in in g p a l e o s t r e s s t e n s o rs f ro m fa u l t p o p u la t i o n s a n d

th e i r s l i p d i r e c t i o n s . T h e s e m e th o d s a re a l l b a s e d u p o n w e l l -k n o w n re l a t i o n s h ip s b e tw e e n

s t r e s s a n d s h e a r a n d u se i t e ra t i v e , n o n - l i n e a r m a th e m a t i c a l a lg o r i t h m s w h ic h s e e k to m in im iz e

th e a n g l e s b e tw e e n t h e c a l c u l a t e d m a x im u m s h e a r s t r e s s d i r e c t i o n a n d th e o b s e rv e d m o v e m e n t

d i re c t io n s o n e a c h f a u l t p l a n e in a p o p u l a t io n . T h e s o l u t io n re t u r n e d i s th e b e s t - f i t p a l e o s t r e s s

te n s o r fo r th e p o p u la t i o n .

B y t a k i n g t h e C o u l o m b f a i l u r e c r i te r i o n i n t o a c c o u n t , s e v e r a l p a l e o s t r e s s a n a l y s i s

p r o g r a m s h a v e b e e n a b l e t o u s e l in e a r , r a t h e r th a n n o n - l i n e a r , m e t h o d s t o s o lv e fo r a p a l e o s t r e s s

t e n s o r . T h e a d v a n ta g e s o f u s in g l i n e a r e q u a t io n s i s th a t th e y a re le ss c o m p u ta t io n a l ly - in t e n s iv e

a n d a re fa r e a s ie r t o s o lv e .

A m a j o r p r o b l e m w i t h c o m p u t a t i o n a l m e t h o d s o f p a l e o s t r e s s a n a l y s i s i s th a t v e r y l i t t l e

w o rk h a s b e e n d o n e o n e v a l u a t in g th e i r e f f e c t i v e n e s s a n d /o r p o s s ib l e l im i t a t i o n s . I f t h e

t e c h n iq u e s r e tu r n re s u l t s c o n s i s t e n t w i th o t h e r m e t h o d s o f e s t im a t in g p a l e o s t r e s s d i re c t i o n s ,

o r w i th v a r io u s k in e m a t i c a n a ly s i s m e th o d s , t h e y a re o f t e n u se d b y g e o lo g i s t s . I f n o t , a n

a t t e m p t m a y b e m a d e to e x p la in w h y , b u t g e o lo g ic a l e x p la n a t io n s a re u s u a l ly so u g h t r a th e r th a n

c r i t i c i z in g th e p a le o s t r e s s a n a ly s i s m e th o d s . T h i s s tu d y i s a n a t t e m p t to fo rm u la te th e p ro b le m

a n d to b e g in s y s te m a t i c a l l y e x a m in in g i t .

F o r m y t h e s i s p r o j e c t , I o b t a i n e d s e v e r a l w o r k i n g v e r s i o n s o f p a l e o s t r e s s a n a l y s i s

c o m p u te r p ro g ra m s . A f t e r m u c h w o rk , I d e c id e d to te s t t w o o f t h e m e th o d s - - t h o s e d e v e lo p e d

b y A n g e l i e r a n d R e c h e s . A r t i f i c i a l fa u l t p o p u l a t i o n s w e r e c r e a t e d f o r t h e s e t e s t s w i t h a s l i p

v e c t o r c a l c u l a t i o n p r o g r a m w h i c h I w r o t e s p e c i f i c a l l y f o r t h a t p u r p o s e . T h e a r t i f i c i a l fa u l t

p o p u la t io n s w e r e c re a t e d u s i n g e x a c t l y th e s a m e i n i t i a l a s s u m p t io n s th a t t h e p a l e o s t r e s s

a n a ly s i s p ro g ra m s u s e d .

A n a r t i f i c i a l fa u l t p o p u l a t i o n i s a s e t o f f a u l t o r i e n t a t i o n s a n d t h e i r a s s o c i a t e d s l i p

d i r e c t io n s c o n s i s te n t w i th a p re d e t e rm in e d s t r e s s f i e l d . F o r a l l o f t h e f a u l t p o p u la t i o n s c r e a t e d ,

th e m o s t c o m p re s s iv e p r in c ip a l s t r e s s a x i s w a s v e r t i c a l w i th a r e la t i v e m a g n i tu d e o f + 1 .0 a n d

th e l e a s t c o m p re s s iv e p r in c ip a l s t r e s s a x i s w a s o r i e n t e d n o r th - s o u th w i th a r e l a t i v e m a g n i tu d e

o f -1 .0 . E n t e r in g th e se p o p u la t i o n s i n t o a p a l e o s t r e s s a n a ly s i s p ro g ra m s h o u ld h a v e ,

th e o re t i c a l l y , r e tu rn e d th e s a m e o r i e n ta t i o n s fo r t h e p r i n c ip a l s t r e s s a x e s .

W i th th i s i n m in d , I c h o se to c re a t e s e v e ra l d i f f e r e n t ty p e s o f a r t i f i c i a l f a u l t p o p u la t i o n s

to t e s t p o s s ib le l im i t a t io n s in p a l e o s t r e s s a n a l y s i s . I u s e d ra n d o m ly - o r i e n te d fa u l t p o p u l a t io n s ,

s p e c ia l - c a s e fa u l t p o p u la t i o n s , a n d f a u l t p o p u la t i o n s w h ic h h a d d a t a a d d e d o r r e m o v e d f ro m

th e m .

T h e r e s u l t s o f th e se t e s t s a re th a t th e tw o p a le o s t r e s s a n a ly s i s p ro g ra m s I e x a m in e d m a y

w o rk s u f f i c i e n t ly w e l l f o r c e r t a in ty p e s o f w e l l - c o n s t r a in e d f a u l t p o p u la t i o n s , b u t o f t e n g iv e

l a r g e e r ro r s w h e n e x a m i n i n g s p e c i a l ty p e s o f f a u l t s e t s s u c h a s c o n j u g a t e f a u l t s , o r t h o r h o m b i c

s y m m e t ry f a u l t s , a n d f a u l t p o p u l a t io n s w h e re a l l o f t h e f a u l t s h a v e v e r y s im i l a r o r i e n t a t io n s .

T h e p a l e o s t r e s s a n a l y s i s p ro g ra m s m a y a l s o b e s e n s i t i v e t o m e a s u re m e n t e r ro r s a n d /o r

e x t r a n e o u s d a ta d e p e n d i n g u p o n s e v e r a l f a c t o r s , i n c lu d in g th e o r i e n t a t i o n s o f th e f a u l t s i n

q u e s t i o n .

In c o n c lu s io n , m u c h m o re w o rk i s c u r re n t ly n e e d e d to f u r th e r e x a m in e th i s t o p i c a n d

t o b e g i n t o f o r m u l a t e g e n e r a l g u i d e l i n e s f o r a p p l y i n g p a l e o s t r e s s a n a l y s i s m e t h o d s t o f a u l t

p o p u la t i o n s g a th e re d b y g e o lo g is t s in th e f i e ld .

E V A L U A T IO N O F C O M P U T A T IO N A L M E T H O D S O F

P A L E O S T R E S S A N A L Y S IS U S IN G F A U L T -S T R IA T IO N D A T A

A t h e s i s p r e s e n t e d t o t h e F a c u l t y

o f th e S t a t e U n i v e r s i ty o f N e w Y o r k

a t A lb a n y

i n p a r t ia l fu l f i l l m e n t o f t h e r e q u i r e m e n t s

fo r th e d e g re e o f

M a s te r o f S c ie n c e

C o l l e g e o f S c ie n c e a n d M a th e m a t i c s

D e p a r tm e n t o f G e o lo g ic a l S c ie n c e s

S t e v e n H e n ry S c h im m r ic h

1 9 9 1

A C K N O W L E D G E M E N T S

I o w e m y g re a t e s t d e b t t o m y th e s i s a d v i s o r , D r . W in th ro p D . M e a n s o f t h e S t a t e

U n iv e r s i ty o f N e w Y o r k a t A lb a n y . W i th o u t h i s c o n s ta n t a s s i s ta n c e , e n c o u ra g e m e n t , a n d

p a t i e n c e t h i s p ro j e c t w o u ld n o t h a v e b e e n p o s s ib l e . I g re a t l y b e n e f i t e d f ro m m y a s so c i a t i o n

w i t h h im .

I a m a l s o in d e b te d to th e fo l l o w in g p e o p le fo r t h e i r a s s i s ta n c e . . .

J a c q u e A n g e l i e r o f th e T e c to n iq u e Q u a n t i t a t i v e , U n iv e rs i t é P ie r r e e t M a r i e C u r i e a t

P a r i s fo r a l l o w in g m e to u s e h i s p a l e o s t r e s s a n a ly s i s p ro g ra m s a n d to C h r i s to p h e r B a r to n o f

L a m o n t - D o h e r t y G e o lo g ic a l O b s e rv a to ry fo r s u p p ly in g m e w i t h c o p ie s o f t h e m .

A rn a u d E t c h e c o p a r o f th e L a b o ra to i r e d e G é o lo g ie S t ru c tu ra le , U n iv e rs i té de s S c i e n c e s

e t T e c h n iq u e s d u L a n g u e d o c , M o n tp e l i e r , F ra n c e fo r a l l o w in g m e t o u s e h i s p a l e o s t r e s s

a n a l y s i s p ro g ra m s a n d to R ic h a rd P l u m b o f S c h l u m b e rg e r D o l l R e s e a rc h in R id g e f i e ld ,

C o n n e c t i c u t f o r s u p p ly in g m e w i t h c o p ie s o f t h e m .

K e n n e th H a rd c a s t l e o f th e D e p a r tm e n t o f G e o lo g y , U n iv e r s i t y o f M a s s a c h u s s e t s a t

A m h e r s t (n o w a t E m e ry a n d G a r re t t G ro u n d w a te r , M e re d i th , N e w H a m p s h i re ) fo r s u p p ly in g

m e w i th c o p i e s o f h i s p a l e o s t r e s s a n a l y s i s p ro g ra m s w h ic h u t i l i z e a lg o r i t h m s d e v e l o p e d b y

Z e 'e v R e c h e s o f t h e D e p a r t m e n t o f G e o lo g y , H e b re w U n iv e rs i t y , Je ru s a le m .

J o h n G e p h a r t o f th e In s t i t u t e fo r th e S t u d y o f C o n t in e n t s , C o rn e l l U n iv e r s i t y fo r

s u p p ly in g m e w i t h c o p ie s o f h i s p a le o s t r e s s a n a ly s i s p ro g ra m .

R i c h a r d A l l m e n d i n g e r o f t h e D e p a r t m e n t o f G e o l o g i c a l S c i e n c e s , C o r n e l l U n i v e r s i ty

fo r su p p l y i n g m e w i th p a l e o s t r e s s a n a l y s e s a n d f i e l d d a t a h e c o l l e c t e d f ro m th e C h i l e a n A n d e s .

S t e v e n W o j t a l o f t h e D e p a r tm e n t o f G e o lo g y , O b e r l i n C o l l e g e , O h io fo r s u p p ly in g m e

w i t h E n g l i s h t r a n s la t i o n s o f s e v e ra l F re n c h p a le o s t r e s s p a p e rs .

W i l l i a m K e l l y o f th e N e w Y o r k S t a t e G e o l o g i c a l S u r v e y i n A l b a n y f o r a l l o w i n g m e t o

ta k e s o m e t i m e o f f to f i n i s h w r i t i n g th i s th e s i s .

T h e n u m e ro u s p e o p le , h e re a t S U N Y A lb a n y a n d e l s e w h e re , w h o g ra c io u s ly g a v e m e

p re p r in t s o f th e i r p a p e r s , e x p la in e d th e i r r e s e a rc h to m e , a n d to o k th e t im e to m a k e h e l p fu l

s u g g e s t io n s a n d c r i t i c i s m s o f m y w o rk .

L a s t , b u t c e r t a in ly n o t l e a s t , I w o u ld l i k e t o t h a n k D e b r a L e n a r d . H e r c o n s ta n t

e n c o u r a g e m e n t ( i . e . " W r i te y o u r d a m n t h e s i s a l r e a d y ! " ) k e p t m e g o i n g a t t i m e s w h e n I w o u l d

r a t h e r h a v e q u i t . S h e p r o v i d e d m e w i t h i n v a l u a b l e a s s i s t a n c e b y h e l p i n g m e t ra n s l a t e

p a le o s t r e s s p ro g ra m d o c u m e n ta t io n f ro m F re n c h in to E n g l i s h , b y b e in g in c re d ib ly su p p o r t i v e ,

a n d b y b e in g o n e o f t h e f e w p e o p le w h o a c tu a l ly l i s t e n e d to m e w h e n I t a l k e d a b o u t p a l e o s t r e s s

a n a ly s i s .

R e s e a rc h l e a d i n g to th i s t h e s i s w a s p a r t i a l l y s u p p o r t e d b y N a t io n a l S c i e n c e F o u n d a t io n

G r a n t E A R - 8 6 0 6 9 6 1 t o W i n t h r o p D . M e a n s , a t w o - y e a r t e a c h i n g a s s i s t a n t s h i p f r o m t h e S t a t e

U n iv e r s i ty o f N e w Y o r k a t A lb a n y , a n d b y o c c a s s io n a l lo a n s o f c a s h (w h ic h , u n fo r tu n a t e ly ,

h a v e y e t t o b e re p a id ) f r o m b o th p a re n ts a n d g ra n d p a re n ts .

I h a v e y e t t o s e e a n y p r o b le m , h o w e ve r c o m p l ic a te d , w h ic h , w h e n lo o k ed a t in th e r ig h t

w a y , d id n o t b e c o m e s t i l l m o re c o m p l ic a te d .

P o u l A n d e r s o n

(T h o rp e , 1 9 6 9 )

T A B L E O F C O N T E N T S

P a g e

A b s t r a c t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i i

A c k n o w le d g e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

T a b le o f C o n te n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v i i

L i s t o f F ig u re s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

L i s t o f T a b le s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x x i

C H A P T E R 1 : IN T R O D U C T IO N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 .1 P u rp o s e o f th i s S tu d y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 .2 S c o p e o f th i s S tu d y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 .3 T e s t in g P ro c e d u re s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 .4 T h e s i s O rg a n iz a t io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

C H A P T E R 2 : P A L E O S T R E S S A N A L Y S IS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 .1 A n d e r s o n ia n F a u l t C la s s i f i c a t io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 .2 B o t t ' s F o rm u la . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3

2 .3 G ra p h i c a l M e t h o d s o f F a u l t -S t r i a t io n P a l e o s t r e s s A n a ly s i s . . . . . . . . . . 1 6

2 .4 R ig h t -D ih e d r a M e t h o d s o f F a u l t -S t r i a t io n A n a ly s i s . . . . . . . . . . . . . . . . 2 4

C H A P T E R 3 : C O M P U T A T IO N A L P A L E O S T R E S S A N A L Y S IS O F F A U L T

P O P U L A T IO N S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 9

3 .1 E a r ly A t t e m p t s a t C o m p u ta t io n a l P a l e o s t r e s s A n a ly s i s . . . . . . . . . . . . . . 3 9

3 .2 E t c h e c o p a r 's M e t h o d o f P a l e o s t r e s s A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . 4 0

3 .3 M ic h a e l ' s M e t h o d o f P a l e o s t r e s s A n a ly s i s . . . . . . . . . . . . . . . . . . . . . . . . . 4 1

3 .4 G e p h a r t a n d F o r s y th ' s M e t h o d o f P a l e o s t r e s s A n a ly s i s . . . . . . . . . . . . . . 4 3

3 .5 R e c e n t T r e n d s i n P a l e o s t r e s s A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4

C H A P T E R 4 : P R O B L E M S IN P A L E O S T R E S S A N A L Y S IS . . . . . . . . . . . . . . . . . . . . . . 4 6

4 .1 M e a s u re m e n t E r ro r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6

4 .2 D e te rm in in g F a u l t S l ip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0

4 .3 F a u l t M o rp h o lo g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3

4 .4 T h e R e l a t i o n s h ip o f S h e a r S t re s s to F a u l t S l ip D i re c t io n s . . . . . . . . . . . . 5 5

4 .5 F a u l t i n g P h a s e D i f f e re n t i a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3

4 .6 D e te rm in in g a P a l e o s t r e s s T e n s o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5

4 .7 D is c u s s io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6

C H A P T E R 5 : G E N E R A T I N G A R T I F IC IA L F A U L T P O P U L A T I O N S . . . . . . . . . . . . . 6 9

5 .1 T h e o ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9

5 .2 D e r iv in g B o t t ' s F o rm u la . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2

5 .3 P ro g ra m In p u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 8

5 .4 P ro g ra m P ro c e d u re s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 9

5 .5 P ro g ra m O u tp u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3

5 .6 C re a t in g F a u l t P o p u la t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4

C H A P T E R 6 : A N G E L IE R 'S M E T H O D O F P A L E O S T R E S S A N A L Y S IS . . . . . . . . . . 9 0

6 .1 P ro g ra m A s s u m p t io n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 0

6 .2 T h e o ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1

6 .3 P ro g ra m In p u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 7

6 .4 P ro g ra m P ro c e d u re s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9

6 .5 P ro g ra m O u tp u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9

6 .6 D is c u s s io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 1

C H A P T E R 7 : R E C H E S ' M E T H O D O F P A L E O S T R E S S A N A L Y S IS . . . . . . . . . . . . . 1 0 2

7 .1 P ro g ra m A s s u m p t io n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 2

7 .2 T h e o ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 3

7 .3 P ro g ra m In p u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 7

7 .4 P ro g ra m P ro c e d u re s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 9

7 .5 P ro g ra m O u tp u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 0

7 .6 D is c u s s io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2

C H A P T E R 8 : P A L E O S T R E S S A N A L Y S IS T E S T D A T A . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4

8 .1 C re a t in g th e A r t i f i c i a l F a u l t P o p u l a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4

8 .2 R a n d o m F a u l t -S l ip P o p u la t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 5

8 .3 C re a t in g S p e c i a l -C a s e F a u l t P o p u l a t io n s . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 3

8 .4 A n d e r s o n i a n C o n ju g a t e F a u l t P o p u l a t io n s . . . . . . . . . . . . . . . . . . . . . . . . 1 3 3

8 .5 O r th o rh o m b ic F a u l t P o p u la t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 0

8 .6 R a d ia l S y m m e t ry F a u l t P o p u l a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 1

8 .7 F a u l t P o p u l a t i o n s o f a S i m i l a r O r i e n t a t i o n . . . . . . . . . . . . . . . . . . . . . . . 1 5 1

8 .8 F in a l F a u l t P o p u la t i o n T e s t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 3

C H A P T E R 9 : T E S T I N G P R O C E D U R E S A N D R E S U L T S . . . . . . . . . . . . . . . . . . . . . . . 1 7 6

9 .1 T e s t in g P ro c e d u re s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 6

9 .2 R a n d o m - P o l e F a u l t P o p u l a t i o n R e s u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 6

9 .3 S p e c i a l -C a s e F a u l t P o p u l a t io n R e s u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 3

9 .4 O th e r T e s t R e s u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 7

C H A P T E R 1 0 : C O N C L U S IO N S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0 0

1 0 .1 W h a t d o th e R e s u l t s M e a n ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0 0

1 0 .2 P ra c t i c a l P ro b le m s in E v a lu a t in g P a l e o s t r e s s A n a ly s i s P ro g ra m s . . . 3 0 1

1 0 .3 S u g g e s t io n s fo r F u tu re W o rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0 6

A P P E N D IX A : G L O S S A R Y O F S Y M B O L S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0 8

A P P E N D IX B : A R T IF IC IA L F A U L T P O P U L A T IO N D A T A . . . . . . . . . . . . . . . . . . . . . 3 1 1

A P P E N D IX C : S L IP V E C T O R C A L C U L A T IO N P R O G R A M . . . . . . . . . . . . . . . . . . . . . 3 2 2

A P P E N D IX D : F A U L T P L A N E P L O T T IN G P R O G R A M . . . . . . . . . . . . . . . . . . . . . . . . . 3 7 4

A P P E N D IX E : V E C T O R A N G L E C A L C U L A T IO N P R O G R A M . . . . . . . . . . . . . . . . . . 3 8 4

R e fe re n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 8 8

L IS T O F F IG U R E S

P a g e

2 -1 A n in f in i t e s im a l p r i s m < o p q rs t> s i t u a t e d w i th in a r i g h t -h a n d e d X Y Z

c o o rd in a t e s y s t e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 - 2 T h e th r e e A n d e r s o n i a n c l a s s e s o f c o n ju g a t e f a u l t s e t s . . . . . . . . . . . . . . . . . . . . . . . . . 1 4

2 -3 A p lo t o f th e s h e a r s t r e s se s a n d n o rm a l s t r e s se s a c t i n g u p o n a p la n e . . . . . . . . . . . . . 1 8

2 - 4 S t e r e o g r a p h i c p r o j e c t i o n s h o w i n g 2 4 c o n j u g a t e n o r m a l fa u l t s a n d t h e i r

a s s o c i a t e d s l i p v e c to r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9

2 -5 S t e re o g ra p h ic p ro je c t io n s h o w in g th e re l a t i o n s h ip b e tw e e n a fa u l t p la n e ,

t h e n o rm a l a n d s l i p v e c to rs , a n d th e m -p la n e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0

2 -6 S t e re o g ra p h i c p ro j e c t io n s h o w in g th e m -p la n e m e th o d o f lo c a t in g a

p r in c ip a l s t r a in a x i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2

2 - 7 A n a ly z in g a c o m m o n in t e r s e c t io n p o i n t (C IP ) o f m -p l a n e s . . . . . . . . . . . . . . . . . . . . 2 3

2 -8 S c h e m a t i c d i a g ra m a n d s t e r e o g ra p h i c p ro j e c t io n i l l u s t r a t i n g th e

re la t i o n s h ip b e tw e e n a n o rm a l f a u l t p la n e , i t s a s s o c ia t e d a u x i l i a ry p l a n e ,

t h e z o n e s o f c o m p re s s io n a n d d i l a t i o n , a n d th e a x e s o f c o m p re s s io n a n d

te n s io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5

2 -9 S t e re o g ra p h ic p ro je c t io n s s h o w in g th e r e l a t i o n s h ip b e tw e e n a fa u l t p la n e ,

i t s a u x i l i a ry p l a n e , a n d i t s m o v e m e n t p la n e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7

12 -1 0 S t e re o g ra p h i c p ro j e c t io n s s h o w in g h o w th e r e g io n c o n ta in in g F i s

c o n s t r a in e d b y n o rm a l f a u l t p o p u la t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8

2 -1 1 S t e re o g ra p h i c p ro j e c t io n s h o w in g 2 0 s l i p v e c to r s r e p re s e n t in g M v a lu e s

ra n g in g f ro m 0 .0 to 1 .0 o n a f a u l t p la n e w i th a n o rm a l v e c to r o r i e n te d a t

7 0 /0 3 0 d e g re e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0

2 -1 2 S t e re o g ra p h i c p ro j e c t io n s h o w in g th e r i g h t d ih e d ra d e f in e d b y L i s le (1 9 8 7 )

a s A a n d B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1

2 - 1 3 S t e r e o g r a p h i c p r o j e c t i o n o f t h r e e f a u l t s w i t h s l i p v e c t o r s S a n d f a u l t

n o rm a l s N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2

1 32 -1 4 S t e re o g ra p h ic p ro je c t io n s h o w in g th e F a n d F f i e l d s c o n s t ru c t e d b y

s u p e r im p o s in g th e f a u l t d a t a f ro m f ig u re 2 -1 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3

2 -1 5 S t e re o g ra p h i c p ro j e c t io n s h o w in g th e r e g io n s c o n s t ru c t e d b y s u p e r im p o s in g

th e A a n d B d ih e d ra f ro m th e f a u l t d a t a i n f i g u re 2 -1 3 . . . . . . . . . . . . . . . . . . . . . . . . . 3 4

1 32 -1 6 S t e re o g ra p h ic p ro je c t io n s h o w in g th e F a n d F f i e l d s c o n s t ru c t e d b y

u t i l i z i n g L i s le ' s (1 9 8 7 ) c o n s t r a in t o n th e f a u l t d a t a f ro m f ig u re 2 -1 3 . . . . . . . . . . . . 3 6

1 32 -1 7 S t e re o g ra p h ic p ro je c t io n s h o w in g th e F a n d F r e g io n s d e r iv e d u s in g

L i s l e ' s (1 9 8 7 ) m e t h o d o n fo u r th r u s t f a u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7

1 32 -1 8 S t e re o g ra p h ic p ro je c t io n s h o w in g th e F a n d F r e g io n s d e r iv e d u s in g

L i s l e 's (1 9 8 7 ) m e th o d o n a c o n ju g a te s e t o f fo u r n o rm a l f a u l t s a n d a n

o r th o rh o m b ic s e t o f fo u r n o rm a l f a u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 8

3 -1 N o r th e a s t q u a d ra n t o f a s t e r e o g ra p h i c p ro j e c t io n s h o w in g th e r e l a t i v e

m a g n i tu d e s o f s h e a r s t r e s s e s o n f a u l t p l a n e s re p re s e n t e d b y t h e i r p o l e s . . . . . . . . . 4 2

4 -1 G ra p h o f th e m a x im u m s t r i k e e r ro r o f a f a u l t p l a n e a s a fu n c t io n o f th e

d ip a n g le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 8

4 - 2 G ra p h o f th e m a x im u m t r e n d e r ro r o f a s l ic k e n l in e o n a f a u l t p l a n e a s a

f u n c t i o n o f t h e p i t c h o f t h e s l i c k e n l i n e a n d t h e d i p a n g l e o f t h e f a u l t . . . . . . . . . . . 4 9

4 -3 D ia g ra m d e m o n s t r a t i n g h o w th e n e t s l i p v e c to r o n a fa u l t p l a n e m a y b e th e

re s u l t o f s e v e r a l d i s t i n c t s l i p e v e n t s w i th d i f f e r in g s l i p d i r e c t i o n s . . . . . . . . . . . . . . 5 1

4 -4 C ro s s -s e c t io n a l v ie w o f a fa u l t p la n e s h o w in g h o w s t e p s m a y b e u se d a s

s e n se -o f - s l i p in d i c a to rs o n fa u l t s u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2

4 -5 D ia g ra m d e m o n s t r a t i n g th e d i f f e r e n c e b e t w e e n a f a u l t ' s s l i p p l a n e a n d t h e

a c tu a l f a u l t s u r f a c e w h ic h m a y n o t b e p la n a r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6

4 -6 T w o fa u l t t y p e s w i th a ro t a t i o n a l c o m p o n e n t o f s l i p . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 8

4 -7 C ro s s -s e c t io n a l v ie w o f a fa u l t s u r f a c e p a ra l l e l t o sh e a r in g w i th a n

a s p e r i t y A c re a t in g a n a n g le N w i th th e f a u l t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 9

4 -8 A n u n d u la t i n g fa u l t p l a n e w i th th e m a x im u m s h e a r s t r e s s d i r e c t i o n m a k in g

a n a n g le " w i th th e l o n g a x i s o f th e u n d u la t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1

4 - 9 G r a p h o f t h e c h a n g e i n p h i (M ) a s a lp h a (" ) v a r i e s f ro m 0 ° t o 9 0 ° . . . . . . . . . . . . . . 6 4

4 -1 0 M o h r c i r c l e s d e m o n s t r a t i n g h o w tw o fa u l t s o f s l i g h t ly d i f f e r e n t

1o r ie n t a t io n s w i l l s l i p a t d i f f e r e n t t im e s a s F i n c re a s e s . . . . . . . . . . . . . . . . . . . . . . . . 6 7

5 - 1 R e l a t i o n s h i p b e t w e e n a p l a n e X Y Z s i t u a t e d w i t h i n a g e o g r a p h i c c o o r d i n a t e

1 2 3s y s t e m a n d t h e p r in c ip a l s t r e s s a x e s F , F , a n d F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 0

5 -2 D e te rm in in g th e s h e a r s t r e s s a n d n o rm a l s t r e s s m a g n i tu d e s a c t i n g u p o n a

fa u l t p l a n e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3

5 - 3 R e l a t i o n s h i p b e t w e e n a p l a n e X Y Z s i t u a t e d i n a g e o g r a p h i c c o o r d i n a t e

1 2 3s y s t e m , th e p r in c ip a l s t r e s s a x e s F , F , a n d F , a n d th e d i r e c t i o n

1 2 3c o s in e s l , l , a n d l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4

5 -4 S t e re o g ra p h ic p ro je c t io n s h o w in g th re e a rb i t r a r i l y o r i e n te d p r in c ip a l

s t r e s s e s in re l a t i o n to a n a rb i t r a r i l y o r i e n te d f a u l t p la n e . . . . . . . . . . . . . . . . . . . . . . . 8 0

5 -5 In te ra c t iv e s c re e n d i s p la y e d b y th e s l i p v e c to r c a l c u la t i o n p ro g ra m a s

th e u s e r e n t e r s th e in i t i a l d a t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1

5 -6 S i m p l i f i e d f l o w c h a r t d e m o n s t r a t i n g th e m a th e m a t i c a l a lg o r i t h m u s e d b y

th e s l i p v e c to r c a l c u l a t i o n p ro g ra m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5

5 -7 G ra p h o f 2 1 M v a lu e s fo r e a c h s l i p v e c to r a s th e y ra n g e f ro m 0 .0 to 1 .0

v e rs u s th e p i t c h o f th e s l i p v e c to r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7

5 -8 S t e re o g ra p h i c p ro j e c t io n o f 2 1 s l i p v e c to r s r e p re s e n t in g v a lu e s ra n g in g

f ro m 0 .0 to 1 .0 o n a f a u l t p l a n e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 8

6 -1 G e o m e t ry o f th e s t r e s se s o n a s t r i a t e d f a u l t p l a n e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2

6 -2 D ia g ra m o f a f a u l t p l a n e l o o k in g d o w n th e p lu n g e o f th e n o rm a l v e c to r

a t t h e fo o tw a l l b lo c k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4

6 - 3 S te r e o g r a p h ic p ro j e c t io n s h o w in g a c o n ju g a t e s e t o f tw o n o rm a l fa u l t s . . . . . . . . 1 0 0

7 - 1 S te r e o g r a p h ic p ro j e c t io n s h o w in g a c o n ju g a t e s e t o f tw o n o rm a l fa u l t s . . . . . . . . 1 0 8

7 - 2 F ig u re s h o w in g th e g r a p h ic a l o u t p u t f ro m R e c h e s ' m e t h o d o f p a l e o s t r e s s

a n a ly s i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1

8 -1 S t e re o g ra p h ic p ro je c t io n o f f a u l t p o p u la t i o n R P -0 1 w h e re M = 0 .0 0 . . . . . . . . . . . 1 1 8

8 -1 0 S t e re o g ra p h ic p ro je c t io n o f f a u l t p o p u la t i o n R P -0 2 w h e re M = 1 .0 0 . . . . . . . . . . . 1 2 7

8 -1 6 S t e re o g ra p h ic p ro je c t io n o f f a u l t p o p u la t i o n A C -0 1 w h e re M = 0 .0 0 . . . . . . . . . . 1 3 5

8 -2 6 S t e re o g ra p h ic p ro je c t io n o f f a u l t p o p u la t i o n O S -0 1 w h e re M = 0 .0 0 . . . . . . . . . . . 1 4 6

8 -3 1 S t e re o g ra p h ic p ro je c t io n o f f a u l t p o p u la t i o n R S -0 1 w h e re M = 0 .0 0 . . . . . . . . . . . 1 5 3

8 -3 6 S t e re o g ra p h ic p ro je c t io n o f f a u l t p o p u la t i o n S O -0 1 w h e re M = 0 .0 0 . . . . . . . . . . . 1 5 8

8 -5 1 S t e re o g ra p h i c p ro j e c t io n o f A n g e l i e r ' s (1 9 7 9 ) f a u l t p o p u la t i o n F D -0 1 . . . . . . . . . 1 7 4

9 - 1 R e c h e s ' m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n R P - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . . . . 1 7 7

9 - 1 0 R e c h e s ' m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n R P - 0 2 a t M = 1 .0 0 . . . . . . . . . . . . . . . . 1 8 6

9 - 1 6 A n g e l i e r ' s m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n R P - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . 1 9 2

9 - 3 1 R e c h e s ' m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n A C - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . . . 2 1 5

9 - 3 6 A n g e l i e r ' s m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n A C - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . 2 2 0

9 - 5 1 R e c h e s ' m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n O S - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . . . 2 3 8

9 - 5 4 R e c h e s ' m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n O S - 0 1 a t M = 0 .7 5 . . . . . . . . . . . . . . . . 2 4 1

9 - 5 6 A n g e l i e r ' s m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n O S - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . 2 4 3

9 - 6 1 R e c h e s ' m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n R S - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . . . 2 5 1

9 - 6 2 R e c h e s ' m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n R S - 0 1 a t M = 0 .2 5 . . . . . . . . . . . . . . . . 2 5 2

9 - 6 6 A n g e l i e r ' s m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n R S - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . 2 5 6

9 - 7 1 R e c h e s ' m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n S O - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . . . 2 6 3

9 - 7 6 A n g e l i e r ' s m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n S O - 0 1 a t M = 0 .0 0 . . . . . . . . . . . . . 2 6 8

9 - 1 0 0 A n g e l i e r ' s m e t h o d r e s u l t s fo r fa u l t p o p u l a t i o n S O - 0 3 a t M = 1 .0 0 . . . . . . . . . . . . . 2 9 4

9 -1 0 1 R e c h e s ' m e th o d re s u l t s fo r f a u l t p o p u la t i o n F D -0 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 8

9 -1 0 2 A n g e l i e r ' s m e th o d re s u l t s fo r f a u l t p o p u la t i o n F D -0 1 . . . . . . . . . . . . . . . . . . . . . . . . 2 9 9

L IS T O F T A B L E S

P a g e

5 - 1 T a b le o f d a t a g e n e r a t e d b y t h e c a l c u l a t io n o f 2 1 s l i p v e c t o r s . . . . . . . . . . . . . . . . . . . 8 6

8 - 1 T a b l e o f d a t a l i s t i n g A n g e l i e r ' s ( 1 9 7 9 ) p o p u l a t i o n o f 3 8 n o r m a l fa u l t s . . . . . . . . . . 1 7 5

9 -1 R e c h e s ' a n d A n g e l i e r ' s r e s u l t s fo r f a u l t p o p u la t i o n R P - 0 1 . . . . . . . . . . . . . . . . . . . . 2 1 0

9 -4 R e c h e s ' a n d A n g e l i e r ' s r e s u l t s fo r f a u l t p o p u la t i o n A C -0 1 . . . . . . . . . . . . . . . . . . . . 2 3 5

9 -5 R e c h e s ' a n d A n g e l i e r ' s r e s u l t s fo r f a u l t p o p u la t i o n A C -0 2 . . . . . . . . . . . . . . . . . . . . 2 3 6

9 -6 R e c h e s ' a n d A n g e l i e r ' s r e s u l t s fo r f a u l t p o p u la t i o n O S - 0 1 . . . . . . . . . . . . . . . . . . . . 2 4 8

9 -7 R e c h e s ' a n d A n g e l i e r ' s r e s u l t s fo r f a u l t p o p u la t i o n R S - 0 1 . . . . . . . . . . . . . . . . . . . . 2 6 1

9 -8 R e c h e s ' a n d A n g e l i e r ' s r e s u l t s fo r f a u l t p o p u la t i o n S O -0 1 . . . . . . . . . . . . . . . . . . . . 2 7 3

9 -9 R e c h e s ' a n d A n g e l i e r ' s r e s u l t s fo r f a u l t p o p u la t i o n S O -0 2 . . . . . . . . . . . . . . . . . . . . 2 9 5

9 -1 0 R e c h e s ' a n d A n g e l i e r ' s r e s u l t s fo r f a u l t p o p u la t i o n S O -0 3 . . . . . . . . . . . . . . . . . . . . 2 9 6

C H A P T E R 1

IN T R O D U C T IO N

A n u m e r ic a l a l g o r i t h m fo r p a l e o s t re s s a n a l y s i s u s i n g f a u l t p o p u l a t io n s w a s f i r s t

p ro p o s e d b y C a re y a n d B ru n ie r (1 9 7 4 ) s ix t e e n y e a r s a g o (A n g e l i e r a n d G o g u e l , 1 9 7 9 ;

E tc h e c o p a r , e t . a l . , 1 9 8 1 ; A n g e l i e r , e t . a l . , 1 9 8 2 ; C é lé r i e r , 1 9 8 8 ; A n g e l i e r , 1 9 8 9 ) . S in c e th a t

t im e , s e v e r a l d i f f e r e n t c o m p u ta t io n a l m e th o d s h a v e b e e n p ro p o s e d (C a re y a n d B ru n ie r , 1 9 7 4 ;

A rm i j o a n d C is te rn a s , 1 9 7 8 ; A n g e l i e r , 1 9 7 9 ; E tc h e c o p a r , e t . a l . , 1 9 8 1 ; A n g e l i e r , e t . a l . , 1 9 8 2 ;

V a s s e u r , e t . a l . , 1 9 8 3 ; A n g e l i e r , 1 9 8 4 ; G e p h a r t a n d F o r s y t h , 1 9 8 4 ; M ic h a e l , 1 9 8 4 ; R e c h e s ,

1 9 8 7 ; A n g e l i e r , 1 9 8 9 ) , e a c h h a v i n g d i s t i n c t a d v a n t a g e s o v e r th e p r e c e d i n g o n e s . T h e s e

n u m e r i c a l m e th o d s a re a l l b a s e d u p o n th e t h e o r e t i c a l r e l a t i o n sh ip s b e tw e e n s t r e s s a n d sh e a r

d e s c r i b e d b y W a l l a c e ( 1 9 5 1 ) a n d B o t t ( 1 9 5 9 ) a n d u s e i t e r a t i v e m e t h o d s t o s e e k a b e s t - f i t

b e tw e e n t h e o b s e rv e d s l i p d i r e c t i o n s o f fa u l t s a n d th e d i r e c t i o n s o f m a x im u m s h e a r s t r e s s o n

e a c h fa u l t p la n e fo r d i f fe re n t p a le o s t r e s s t e n s o rs (E tc h e c o p a r , e t . a l . , 1 9 8 1 ; A n g e l i e r , 1 9 8 4 ;

G e p h a r t a n d F o r s y t h , 1 9 8 4 ; M i c h a e l , 1 9 8 4 ; R e c h e s , 1 9 8 7 ; A n g e l i e r , 1 9 8 9 ) .

P a l e o s t r e s s a n a ly s i s p ro g ra m s r e q u i r e a n i n i t i a l d a t a s e t o f f a u l t s a s su m e d , o r k n o w n ,

t o h a v e b e e n a c t i v a t e d d u r i n g a s i n g l e t e c t o n i c e v e n t w i t h i n a h o m o g e n e o u s s t r e s s f i e l d . A f a u l t

d a tu m c o n s i s t s o f t h e f a u l t ' s o r i e n t a t i o n , s l i p d i re c t i o n , a n d s e n s e o f s l i p . T h i s f a u l t p o p u la t i o n

1 2 3d a ta i s u s e d t o c a l c u l a t e t h e o r i e n t a t io n o f th e th r e e p r in c ip a l s t r e s s a x e s F , F , a n d F a n d a

v a lu e s ig n i fy i n g s o m e r a t io o f th e i r r e l a t iv e m a g n i tu d e s , c o m m o n ly d e n o t e d a s M ( A n g e l i e r ,

1 9 7 9 ; M i c h a e l , 1 9 8 4 ) .

1 .1 P u r p o se o f th is S tu d y

P a l e o s t r e s s a n a l y s i s t e c h n i q u e s h a v e r e c e n t l y b e e n a p p l i e d b y m a n y g e o l o g i s t s t o

reg ion a l f a u l t p o p u la t i o n s (A n g e l i e r a n d M e c h le r , 1 9 7 7 ; A n g e l i e r , 1 9 7 9 ; A n g e l i e r a n d G o g u e l ,

1 9 7 9 ; A y d in , 1 9 8 0 ; E tc h e c o p a r , e t . a l . , 1 9 8 1 ; A n g e l i e r , e t . a l . , 1 9 8 2 ; A n g e l i e r , 1 9 8 4 ;

M i c h a e l , 1 9 8 4 ; A le k s a n d ro w s k i , 1 9 8 5 ; A n g e l i e r , e t . a l . , 1 9 8 5 ; F r i z z e l l a n d Z o b a c k , 1 9 8 7 ;

H a n c o c k , e t . a l . , 1 9 8 7 ; Ju l i e n a n d C o rn e t , 1 9 8 7 ; L i s le , 1 9 8 7 ; P f i f f n e r a n d B u rk h a rd , 1 9 8 7 ;

R e c h e s , 1 9 8 7 ; S a s s i a n d C a re y -G a i lh a rd i s , 1 9 8 7 ; C a p u to a n d C a p u to , 1 9 8 8 ; C é l é r i e r , 1 9 8 8 ;

L a r ro q u e a n d L a u re n t , 1 9 8 8 ; L i s le , 1 9 8 8 ; A n g e l i e r , 1 9 8 9 ; H a rd c a s t l e , 1 9 8 9 ; H a t z o r a n d

R e c h e s , 1 9 8 9 ; M a n n i n g a n d d e B o e r , 1 9 8 9 ; W a l l b r e c h e r a n d F r i tz , 1 9 8 9 ; U m h o e f e r , 1 9 9 0 ) in

o r d e r to d e r i v e r e g i o n a l p a l e o s t r e s s t e n s o r s . A c l o s e e x a m i n a t i o n o f p u b l i s h e d s t u d i e s i n d i c a t e

t h a t n u m e r i c a l p a l e o s t r e s s a n a l y s i s t e c h n i q u e s a r e c o m m o n l y a p p l i e d t o f a u l t p o p u l a t i o n s o n l y

w h e n th e r e s u l t s a r e c o n s i s te n t w i th o th e r m e th o d s o f e s t im a t in g p a l e o s t r e s se s (F r iz z e l l a n d

Z o b a c k , 1 9 8 7 ; H a n c o c k , e t . a l . , 1 9 8 7 ; P f i f f n e r a n d B u r k h a r d , 1 9 8 7 ; L a r r o q u e a n d L a u r e n t ,

1 9 8 8 ; H a rd c a s t l e , 1 9 8 9 ; M a n n in g a n d d e B o e r , 1 9 8 9 ; W a l lb re c h e r a n d F r i t z , 1 9 8 9 ; H a t z o r

a n d R e c h e s , 1 9 9 0 ) . U n f o r t u n a t e l y , v e r y l i t t l e w o r k h a s b e e n p u b l i s h e d o n t h e p o s s i b l e

l im i ta t ion s o f i n d iv id u a l p a le o s t r e s s a n a ly s i s p ro g ra m s c u r r e n t l y in u s e (A n g e l i e r , e t . a l . , 1 9 8 2 ;

A n g e l i e r , 1 9 8 4 ; C é lé r i e r , 1 9 8 8 ) a n d I a m n o t a w a re o f a n y p u b l i s h e d p a p e rs p ro v i d i n g a

d e ta i l e d s tu d y o f th e l im i t a t i o n s o f p a le o s t r e s s a n a ly s i s p ro g ra m s in g e n e ra l . T h i s im p l i e s th a t

s o m e re s e a rc h e r s m a y b e a p p ly in g p a l e o s t r e s s a n a ly s i s t o r e g io n a l f a u l t p o p u la t i o n s w i th o u t

h a v in g a c l e a r id e a o f h o w a p p ro p r i a t e th e a p p l i c a t i o n o f s u c h t e c h n iq u e s t o t h e i r d a ta s e t s m a y

b e (E d e lm a n , 1 9 8 9 ) a l t h o u g h , s in c e w o rk o n th i s t h e s i s h a s s t a r t e d , s e v e r a l r e s e a rc h e r s h a v e

b e g u n to e x a m in e th i s im p o r t a n t t o p ic (P e rs h in g , 1 9 8 9 ; P o l l a rd , 1 9 9 0 ) .

M y s tu d y i s a n a t t e m p t to c o r re c t th i s p ro b le m b y sy s te m a tic a l ly e x a m in in g tw o w id e ly -

u s e d p a le o s t r e s s a n a ly s i s p ro g ra m s a n d d e m o n s t r a t i n g th a t t h e y b o th p o s s e s s im p o r t a n t

l im i t a t io n s . G e o lo g i s t s s h o u ld b e m a d e a w a r e o f t h e s e l im i t a t io n s b e fo r e t h e y u s e p a l e o s t r e s s

a n a ly s i s t e c h n iq u e s to d e te rm in e a p o s s ib le p a le o s t r e s s t e n so r f ro m f i e ld d a ta .

1 .2 S c o p e o f th is S tu d y

T h e tw o p a l e o s t r e s s a n a l y s i s p r o g r a m s I c h o s e to e x a m in e fo r th i s s t u d y w e r e t h o s e

d e v e lo p e d b y A n g e l i e r ( A n g e l i e r , 1 9 7 9 ; A n g e l i e r , 1 9 8 9 ) a n d R e c h e s (R e c h e s , 1 9 8 7 ) . T h e s e

p r o g r a m s w e r e c h o s e n fo r t e s t in g b e c a u s e t h e l i t e r a t u re i n d i c a t e d th a t t h e y a re t h e m o s t

c o m m o n ly u s e d p a le o s t r e s s a n a ly s i s p ro g ra m s (A n g e l i e r , 1 9 7 9 ; A n g e l i e r , e t . a l . , 1 9 8 2 ;

A n g e l i e r , 1 9 8 4 ; A n g e l i e r , e t . a l . , 1 9 8 5 ; P f i f f n e r a n d B u r k h a r d , 1 9 8 7 ; R e c h e s , 1 9 8 7 ; A n g e l i e r ,

1 9 8 9 ; H a r d c a s t l e , 1 9 8 9 ; H a tz o r a n d R e c h e s , 1 9 9 0 ) . I h a v e a l s o b e g u n e x a m in in g o th e r

p a le o s t r e s s a n a ly s i s p ro g ra m s i n c lu d in g t h o s e d e v e lo p e d b y E tc h e c o p a r (E tc h e c o p a r , e t . a l . ,

1 9 8 1 ) , M ic h a e l (M ic h a e l , 1 9 8 4 ) , G e p h a r t (G e p h a r t a n d F o r sy th , 1 9 8 4 ) , a n d L i s le (L i s le , , 1 9 8 8 )

a l th o u g h th e r e s u l t s f ro m th o s e p ro g ra m s a r e t o o p re l im in a ry to b e in c lu d e d h e re a n d w i l l b e

a d d r e s s e d i n a f u t u r e p a p e r .

T o e v a lu a t e t h e u s e f u l n e s s o f t h e p a l e o s t r e s s a n a l y s i s p ro g ra m s t e s t e d , I w i l l a d d r e s s

th e fo l l o w in g th re e q u e s t i o n s in th i s th e s i s .

1 . D o th e tw o p a l e o s t r e s s a n a ly s i s p ro g ra m s c h o s e n fo r t e s t i n g h a v e s ig n i f i c a n t

l im i t a t io n s ?

2 . I f s o , w h a t e x a c t ly a re t h o s e l im i t a t io n s ?

3 . H o w d o e s t h i s i n f o r m a t i o n p e r t a i n t o t h e g e o l o g i s t a p p l y i n g p a l e o s t r e s s a n a l y s i s

t e c h n iq u e s ?

S e v e ra l r e s e a rc h e r s h a v e s h o w n th a t e a c h o f t h e p a le o s t r e s s a n a ly s i s p ro g ra m s I t e s t e d

w i l l y i e l d g e o l o g ic a l ly - re a s o n a b l e , o r e x p e c t e d , r e s u l t s fo r c e r t a i n f a u l t p o p u l a t i o n s ( A n g e l i e r ,

1 9 7 9 ; A n g e l i e r , e t . a l . , 1 9 8 2 ; A n g e l i e r , 1 9 8 4 ; A n g e l i e r , e t . a l . , 1 9 8 5 ; P f i f fn e r a n d B u rk h a rd ,

1 9 8 7 ; R e c h e s , 1 9 8 7 ; A n g e l i e r , 1 9 8 9 ; H a rd c a s t l e , 1 9 8 9 ; H a tz o r a n d R e c h e s , 1 9 9 0 ) . I

s p e c i f i c a l l y s e t o u t to f in d s i t u a t io n s in w h i c h th e m e t h o d s w o u ld n o t y i e l d th e e x p e c t e d re s u l t s .

In t h i s w a y , I h o p e d to d i s c o v e r a n d e v a lu a t e a n y w e a k n e s s e s in h e r e n t in t h e s e p r o g r a m s .

T h e h y p o t h e s i s I w i s h t o p r o v e i n t h i s t h e s i s i s t h a t c o m p u ta t io n a l m e th o d s o f

p a le o s tre ss a n a ly s is h a v e se v era l s ig n i f ic a n t l im i ta t io n s th a t g e o lo g is t s m u s t b e a w a r e o f w h e n

u s in g th e se te ch n iq u e s to d e r iv e a re g io n a l p a le o s tre ss te n so r f ro m n a tu ra l fa u l t p o p u la t io n s .

1 .3 T e st in g P r o c e d u r e s

A l l o f th e p a le o s t r e s s a n a ly s i s p ro g ra m s t e s t e d m a k e th re e v e ry im p o r t a n t in i t i a l

a s s u m p t io n s .

1 . T h e f a u l t a n d s l i p o r i e n t a t i o n s w h ic h c o m p r i s e th e d a t a s e t a r e a s so c i a t e d w i th a

u n i q u e , h o m o g e n e o u s r e g i o n a l p a l e o s t r e s s t e n s o r .

2 . T h e f a u l t s e a c h b e h a v e in d e p e n d e n t ly o f o n e a n o th e r a n d d o n o t in t e r a c t m e c h a n i c a l ly .

3 . T h e m o v e m e n t v e c to r fo r e a c h f a u l t p l a n e c o r re s p o n d s to th e d i r e c t i o n o f m a x im u m

s h e a r s t r e s s w i th in th a t p la n e .

A r t i f i c i a l f a u l t p o p u l a t i o n s c o n s i s te n t w i th a k n o w n s t r e s s t e n s o r w e re d e r iv e d u s in g

th e se th re e s im p l i fy in g a s s u m p t io n s . T h e s e a r t i f i c i a l f a u l t p o p u la t i o n d a ta s e t s w e re th e n u se d

to t e s t t h e p a l e o s t r e s s a n a l y s i s p r o g r a m s in f o u r d i f f e r e n t w a y s .

1 . T h e a c c u r a c y o f e a c h p a l e o s t r e s s a n a l y s i s p r o g r a m w a s te s te d b y c re a t i n g r a n d o m f a u l t -

s l i p p o p u l a t i o n s c o n s i s t e n t w i t h a c h o s e n s t r e s s t e n s o r . T h e o r i e n t a t i o n s o f th e f a u l t

p l a n e s w e re r a n d o m ly c h o s e n a n d th e i r s l i p d i r e c t i o n s w e re c o in c id e n t w i th th e

d i r e c t i o n o f m a x im u m s h e a r s t r e s s w i th in e a c h p l a n e . T h e s e fa u l t p o p u la t i o n s w e r e

u s e d a s i n p u t fo r th e p a l e o s t r e s s a n a l y s i s p r o g r a m s a n d t h e r e s u l t s w e r e c o m p a r e d t o

th e o r ig in a l s t r e s s t e n s o r u n d e r w h ic h th e f a u l t - s l i p d a t a w e re c r e a t e d .

2 . T h e b e h a v i o r o f e a c h p a l e o s t r e s s a n a l y s i s p r o g r a m w h e n a p p l i e d t o s p e c i a l - c a s e f a u l t

p o p u la t i o n s w a s te s te d b y u s in g th e fo l l o w in g fa u l t p o p u la t i o n s a s so c i a t e d w i th a

k n o w n s t r e s s t e n s o r .

A . S i m p l e A n d e r s o n i a n c o n j u g a t e f a u l t s e t s ( A n d e r s o n , 1 9 5 1 ) .

B . O r th o rh o m b ic , o r rh o m b o h e d ra l , f a u l t p o p u la t i o n s (A y d in a n d R e c h e s , 1 9 8 2 ;

K r a n t z , 1 9 8 6 ; K r a n t z , 1 9 8 9 ) .

C . F a u l t p o p u la t i o n s w h e re a l l f a u l t s h a v e a p p ro x im a te ly th e s a m e o r i e n t a t i o n .

D . F a u l t p o p u la t i o n s w h e re s o m e o r a l l o f th e f a u l t s h a v e a p p ro x im a te ly th e s a m e

o r ie n t a t io n a s t h e p r in c ip a l s t r e s s p l a n e s .

3 . T h e s t a b i l i t y o f t h e c a lc u la t e d p a le o s t r e s s t e n so rs to in su f f i c i e n t d a t a , e x t r a n e o u s d a t a ,

a n d m e a s u re m e n t e r ro r s w a s te s te d b y a p p ly in g th e fo l l o w in g p ro c ed u re s a n d n o t in g th e

e f fe c t u p o n t h e c a l c u l a t e d p a l e o s t r e s s t e n s o r .

A . R a n d o m ly re m o v in g o n e o r m o re f a u l t p l a n e s f ro m a fa u l t p o p u la t i o n .

B . R a n d o m l y a d d i n g o n e o r m o r e f a u l t p l a n e s , w i t h r a n d o m l y c h o s e n s l i p

d i r e c t io n s , t o th e f a u l t p o p u la t i o n .

C . G i v i n g a ± 5 ° v a r i a b i l i t y t o t h e o r i e n t a t i o n s o f th e f a u l t p l a n e n o r m a l a n d s l i p

v e c t o r s .

4 . F i n a l ly , t h e p ro g ra m s w e re c o m p a re d to o n e a n o th e r , u s in g b o th n a tu ra l a n d a r t i f i c i a l

f a u l t p o p u l a t i o n s , t o s e e h o w c o n s i s t e n t t h e r e s u l t s w e r e g i v e n t h e s a m e i n i t i a l d a t a

s e t s . S i n c e o n e a s s u m p t io n s h a re d b y a l l o f t h e p a l e o s t r e s s a n a ly s i s p ro g ra m s i s t h a t

th e re e x i s t s a u n iq u e r e g io n a l p a l e o s t r e s s t e n s o r fo r a n y g iv e n f a u l t p o p u la t i o n a r i s in g

f ro m a s in g l e t e c t o n ic e v e n t , i n c o n s i s te n c i e s b e tw e e n th e o u tp u t o f t h e v a r io u s

p ro g ra m s w o u ld in d ic a te th e i r u n re l i a b i l i t y g iv e n c e r t a in in i t i a l d a ta s e t s .

1 .4 T h e s is O r g a n iz a t io n

I h a v e o rg a n i z e d t h i s t h e s i s i n to fo u r m a in s e c t io n s . C h a p t e r s 2 a n d 3 in t ro d u c e t h e

c o n c e p t o f p a l e o s t r e s s a n a ly s i s a s i t i s a p p l i e d t o f a u l t p o p u la t i o n s a n d g iv e a r e v i e w o f

p r e v i o u s w o r k . T h e s e c o n d s e c t i o n c o n s i s t s o f c h a p t e r 4 w h i c h d i s c u s s e s p r o b l e m s i n h e r e n t in

p a l e o s t r e s s a n a l y s i s , c h a p t e r 5 w h i c h d e s c r i b e s t h e p r o g r a m a n d m e t h o d o l o g y u s e d t o g e n e r a t e

a r t i f i c i a l fa u l t p o p u l a t i o n s , a n d c h a p t e r s 6 a n d 7 w h i c h d e s c r i b e t h e t w o p a l e o s t r e s s a n a l y s i s

p ro g ra m s t e s te d . T h e th i rd s e c t io n , c o n s i s t i n g o f c h a p te r s 8 a n d 9 , i s th e m a in b o d y o f th e

th e s i s a n d d e t a i l s t h e p ro c e d u r e s u se d to t e s t t h e p a l e o s t r e s s a n a ly s i s p ro g ra m s a n d p re s e n t s t h e

re s u l t s o f th o s e te s t s . F in a l ly , c h a p t e r 1 0 p re s e n t s t h e c o n c l u s io n s o f th i s s tu d y a n d s u g g e s t io n s

fo r fu r t h e r w o rk .

C H A P T E R 2

P A L E O S T R E S S A N A L Y S IS

P a le o s t r e s s a n a ly s i s r e fe r s to v a r io u s m e th o d s w h ic h a t t e m p t to d e te rm in e a re g io n a l

s t r e s s t e n s o r c o n s i s te n t w i th e x i s t i n g g e o l o g ic s t ru c tu re s . S e v e r a l d i f f e r e n t t e c h n i q u e s fo r

e s t im a t in g s t r e s s t e n s o rs h a v e b e e n p ro p o s e d . P r in c ip a l s t r e s s d i r e c t i o n s a n d r e l a t i v e

m a g n i tu d e s h a v e b e e n d e t e rm in e d f ro m fa u lt p o p u la t io n s (A n g e l i e r a n d M e c h l e r , 1 9 7 7 ;

A n g e l i e r , 1 9 7 9 ; A n g e l i e r a n d G o g u e l , 1 9 7 9 ; A y d in , 1 9 8 0 ; E tc h e c o p a r , e t . a l . , 1 9 8 1 ; A n g e l i e r ,

e t . a l . , 1 9 8 2 ; A n g e l i e r , 1 9 8 4 ; M ic h a e l , 1 9 8 4 ; A le s a n d ro w s k i , 1 9 8 5 ; A n g e l i e r , e t . a l . , 1 9 8 5 ;

F r i z z e l l a n d Z o b a c k , 1 9 8 7 ; H a n c o c k , e t . a t . , 1 9 8 7 ; L i s le , 1 9 8 7 ; P f i f fn e r a n d B u rk h a rd , 1 9 8 7 ;

R e c h e s , 1 9 8 7 ; S a s s i a n d C a re y -G a i lh a rd i s , 1 9 8 7 ; C a p u to a n d C a p u to , 1 9 8 8 ; C é l é r i e r , 1 9 8 8 ;

L a r ro q u e a n d L a u re n t , 1 9 8 8 ; L i s le , 1 9 8 8 ; A n g e l i e r , 1 9 8 9 ; H a rd c a s t l e , 1 9 8 9 ; M a n n in g a n d d e

B o e r , 1 9 8 9 ; M c B r id e , 1 9 8 9 ; W a l lb re c h e r a n d F r i t z , 1 9 8 9 ; H a t z o r a n d R e c h e s , 1 9 9 0 ;

U m h o e fe r , 1 9 9 0 ) , e a r th q u a ke fo c a l m e c h a n ism d a t a ( M c K e n z i e , 1 9 6 9 ; E l l s w o r th a n d

Z h o n g h u a i , 1 9 8 0 ; V a s s e u r , e t . a l . , 1 9 8 3 ; G e p h a r t a n d F o r s y th , 1 9 8 4 ; Ju l i e n a n d C o rn e t , 1 9 8 7 ;

P f i f fn e r a n d B u r k h a rd , 1 9 8 7 ; W a h l s t r o m , 1 9 8 7 ; M ic h a e l , 1 9 8 7 a ; M ic h a e l , 1 9 8 7 b ; J o n e s ,

1 9 8 8 ) , b o r e h o le e l o n g a t io n d a t a ( Z o b a c k a n d Z o b a c k , 1 9 8 0 ; P l u m b a n d C o x , 1 9 8 6 ; S u t e r ,

1 9 8 6 ; H a n s e n a n d M o u n t , 1 9 9 0 ) , j o i n t se t s (P r ic e , 1 9 6 6 ; E n g e ld e r a n d G e i s e r , 1 9 8 0 ;

E n g e ld e r , 1 9 8 2 ; H a n c o c k , 1 9 8 5 ; H a n c o c k , e t . a l . , 1 9 8 7 ) , d i k e se t s (B e rg e r , 1 9 7 1 ; M u l l e r a n d

P o l la rd , 1 9 7 7 ; D a v id s o n a n d P a rk , 1 9 7 8 ; B o r ra d a i l e , 1 9 8 6 ; R ic e , 1 9 8 6 ; L i s le , 1 9 8 9 ; M a n n in g

a n d d e B o e r , 1 9 8 9 ; H a n s e n a n d M o u n t , 1 9 9 0 ) , c a lc it e e - tw in s (S p a n g , 1 9 7 2 ; S p a n g a n d V a n

D e r L e e , 1 9 7 5 ; L a u re n t , e t . a l . , 1 9 8 1 ; P f i f fn e r a n d B u rk h a rd , 1 9 8 7 ; L a r ro q u e a n d L a u re n t ,

1 9 8 8 ) , v a r io u s m ic r o s tr u c tu r a l f e a t u r es (F r i e d m a n , 1 9 6 4 ; S c o t t , e t . a l . , 1 9 6 5 ; C a r t e r a n d

R a le ig h , 1 9 6 9 ; S p a n g a n d V a n D e r L e e , 1 9 7 5 ; W h i t e , 1 9 7 9 ; P lu m b , e t . a l . , 1 9 8 4 ; P ê c h e r , e t .

a l . , 1 9 8 5 ; L e s p in a s s e a n d P ê c h e r , 1 9 8 6 ; K o w a l l i s , e t . a l . , 1 9 8 7 ; J a n g , e t . a l . , 1 9 8 9 ; L a u b a c h ,

1 9 8 9 ; S h e p a rd , 1 9 9 0 ) , fo ld s (D ie te r i c h a n d C a r t e r , 1 9 6 9 ; M ic h a e l , 1 9 8 4 ) , s ty lo l i t e s

(A r t h a u d a n d M a t t a e u r , 1 9 6 9 ; B u c h n e r , 1 9 8 1 ; H a n c o c k , e t . a l . , 1 9 8 7 ) , kin k b a n d s (G a y a n d

W e i s s , 1 9 7 4 ) , a n d f r a c tu r e m a r k in g s o n jo in t su r fa c e s (B a h a t a n d R a b in o v ic h , 1 9 8 8 ) .

F a u l t - s t r i a t io n p a l e o s t r e s s a n a l y s i s , t h e t o p i c o f t h i s t h e s i s , i s th e s u b s e t o f p a l e o s t r e s s

a n a ly s i s w h ic h a t t e m p ts to e s t im a te th e r e l a t i v e m a g n i t u d e s a n d o r i e n ta t i o n s o f th e t h re e

1 2 3p r in c ip a l s t r e s s e s F , F , a n d F ( m o s t c o m p re s s iv e to l e a s t c o m p re s s iv e r e s p e c t iv e ly ) f ro m

fa u l t p o p u l a t io n s a n d th e i r a s s o c i a t e d s l i p d i r e c t i o n s .

2 .1 A n d er so n ia n F a u lt C la ss i f ic a t io n

T o d e te rm in e th e s t r e s s t e n so r a s s o c ia t e d w i th a d i s p la c e m e n t a l o n g a f a u l t i n a g iv e n

s l i p d i r e c t i o n , s o m e h y p o t h e s i s m u s t b e m a d e a b o u t th e f a i l u r e m e c h a n i s m s in v o lv e d . T h e f i r s t ,

a n d s im p le s t , h y p o th e s i s i s t h a t f a i l u re o c c u r re d w i th in in t a c t i s o t ro p i c ro c k . I n A n d e r s o n 's

c l a s s i c w o rk o n f a u l t i n g ( A n d e r s o n , 1 9 5 1 , p . 7 ) , C o u lo m b 's f a i l u re c r i t e r io n (C o u lo m b , 1 7 7 6 ;

1 2H a n d i n , 1 9 6 9 ) w a s u s e d to p re d i c t t h e o r i e n t a t io n s o f t h e t h r e e p r in c ip a l s t r e s s a x e s F , F , a n d

3F r e s u l t in g in th e t h re e c o m m o n ty p e s o f c o n j u g a t e f a u l t s y s te m s - - t h ru s t , n o rm a l , a n d

w re n c h .

A n d e rs o n b e g a n b y e x a m in in g a n in f in i t e s im a l p r i s m (o r a t l e a s t o n e sm a l l e n o u g h s u c h

t h a t th e s t r e s s e s p r e s e n t a r e h o m o g e n e o u s t h r o u g h o u t i t s v o l u m e ) s i tu a t e d w i t h i n a r ig h t -

1 2 3h a n d e d X Y Z c o o r d i n a t e s y s t e m w h e re F , F , a n d F a r e p a ra l l e l t h e X , Y , a n d Z d i r e c t i o n s

re s p e c t iv e ly ( f i g u re 2 -1 ) . I f f a c e A i s a s su m e d to b e o f u n i t a r e a a n d 2 i s d e f i n e d a s t h e a n g l e

b e tw e e n p l a n e A a n d t h e X -a x i s , t h i s w o u ld i m p ly a n a re a o f s in (2) fo r f a c e < o p r t> a n d a n a re a

o f c o s (2) fo r f a c e < p q r s > . I f t h e s y s t e m i s in e q u i l i b r iu m , th e n a s im p le fo rc e b a l a n c e s h o w s

nt h a t t h e fo r c e a c t i n g n o r m a l to p l a n e A ( w h ic h i s e q u iv a l e n t to t h e n o r m a l s t r e s s F u p o n th e

p l a n e ) i s

F ig u r e 2 - 1 - A n i n f i n i t e s i m a l p r i s m < o p q r s t > s i t u a t e d w i t h i n a r ig h t - h a n d e d X Y Z c o o r d i n a t e

1 2 3s y s t e m . T h e th r e e p r in c ip a l s t r e s s e s F , F , a n d F p a r a l l e l t h e X , Y , a n d Z - a x e s r e s p e c t i v e l y

a n d 2 d e f in e s th e a n g le b e tw e e n p la n e A a n d t h e X -a x is . P l a n e A i s d e f in e d to b e o f u n i t a re a .

n 1 3F = F s in (2) s in (2) + F c o s (2) c o s (2) (1 )

sa n d th e s h e a r s t r e s s F , w h ic h i s fo u n d b y re s o lv in g th e fo rc e s p a ra l l e l t o th e d i r e c t i o n < o q > ,

s 1 3F = F s in (2) c o s (2) - F s in (2) c o s (2) (2 )

w h ic h c a n b e r e d u c e d u s in g th e t r i g o n o m e t r i c i d e n t i t y 2 s in (2) c o s (2) = s i n ( 22) to y i e l d

s 1 3F = [ (F - F ) / 2 ] s i n ( 22) (3 )

I t i s c l e a r , fo r a n y g i v e n m a g n i tu d e s o f p r in c ip a l s t r e s s e s p re s e n t , t h a t t h e s h e a r s t r e s s

w i l l b e g r e a t e s t w h e n s i n ( 22) = ± 1 , o r 2 = ± 4 5 ° . T h e re a r e th u s tw o p l a n e s a t a n y p o in t a c r o s s

w h ic h th e s h e a r s t r e s s m a g n i tu d e s a re a t a m a x im u m .

A n d e r s o n n o te d , h o w e v e r , t h a t i n n a tu ra l a n d a r t i f i c i a l c o n ju g a t e f a u l t s , a n a c u t e a n d

1a n o b tu s e a n g le a r e p re s e n t w i th th e a c u t e a n g l e b i s e c te d b y F a n d th e o b tu se a n g le b i s e c te d

3b y F ( i . e . t h e p r in c ip a l s t r e s s a x e s a r e n o t o r ie n t e d a t 4 5 ° f ro m t h e f a u l t p l a n e s a s e q u a t io n (3 )

p re d i c t s ) . T h i s p r in c ip a l i s o c c a s io n a l l y r e f e r re d to a s H a r tm a n n 's R u le (H a r tm a n n , 1 8 9 6 ;

D e n n i s , 1 9 8 7 , p . 2 3 8 ) a f t e r th e F r e n c h m e t a l l u r g i s t w h o f i r s t f o r m u l a t e d i t . T h e s o l u t i o n t o t h i s

d i s c r e p a n c y b e t w e e n t h e o r y a n d o b s e r v a t i o n i s to t a k e i n t o a c c o u n t th e C o u l o m b f a i l u r e

c r i t e r io n

s 0 nF = C + :F (4 )

0w h e re C i s a c o n s ta n t d e n o t in g th e c o h e s io n o f th e f a u l t s u r f a c e a n d : i s t h e c o e f f i c i e n t o f

in te rn a l f r i c t i o n (C o u lo m b , 1 7 7 6 ; H a n d in , 1 9 6 9 ) . T h i s i s n e e d e d be c a u se f r i c t i o n e x i s t s o n re a l

f a u l t s u r fa c e s . S u b s t i t u t i n g i n t h e n o r m a l s t re s s f r o m e q u a t i o n ( 1 ) re s u l t s in

s 0 1 3F = C + : [F s i n ( 2) + F c o s ( 2) ] (5 )2 2

1 1 3 1 3 3 1 3 1a n d s u b s t i t u t in g th e id e n t i t i e s F / [ (F + F ) / 2 ] + [ (F - F ) / 2 ] a n d F / [ (F + F ) / 2 ] - [ (F -

3F ) / 2 ] i n to e q u a t io n (5 ) a n d re d u c in g i t t h ro u g h s u i t a b l e a lg e b ra i c m a n ip u la t i o n s a n d th e tw o

t r ig o n o m e t r ic i d e n t i t i e s [ s i n (2) + c o s (2) ] = 1 a n d [c o s (2) - s i n (2) ] = c o s ( 22) y i e ld s2 2 2 2

s 0 1 3 1 3F = C + : [ ( (F + F ) / 2 ) - ( (F - F ) / 2 ) c o s ( 22) ] (6 )

1 3 0 1 3 1 3[ (F - F ) / 2 ] s i n ( 22) = C + : [ ( (F + F ) / 2 ) ( (F - F ) / 2 ) c o s ( 22) ] (7 )

T h e p ro b le m i s n o w t o f i n d t h e p la n e s a lo n g w h ic h th e s h e a r in g s t r e s s (w h ic h d r iv e s

s l i p ) w i l l m o s t l i k e ly o v e rc o m e th e n o rm a l s t r e s s (w h ic h a c t s t o r e t a rd s l i p ) . S in c e th e s h e a r

s t r e s s m a y h a v e b o th p o s i t i v e a n d n e g a t iv e v a lu e s , t h e p ro b le m re d u c e s to f i n d in g th e a n g le s

fo r w h ic h

1 3 1 3 1 3[ (F - F ) / 2 ] s i n ( 22) + : [ ( (F + F ) / 2 ) ( (F - F ) / 2 ) c o s ( 22) ] (8 )

i s a t a m a x im u m , a n d

1 3 1 3 1 3[ (F - F ) / 2 ] s i n ( 22) - : [ ( (F + F ) / 2 ) ( (F - F ) / 2 ) c o s ( 22) ] (9 )

i s a t a m in im u m . D i f f e r e n t i a t i n g th e s e tw o e q u a t io n s w i th r e s p e c t to 2 a n d se t t i n g th e m e q u a l

to z e ro y ie ld s

1 3 1 3[ (F - F ) / 2 ] c o s ( 22) ± : [ (F - F ) / 2 ] s i n ( 22) = 0 (1 0 )

w h i c h r e d u c e s t o

c o s ( 22) ± :s i n ( 22) = 0 (1 1 )

o r , a l t e r n a t i v e l y

t a n ( 22) = ± ( 1 / : ) (1 2 )

S i n c e t h e c o e f f i c i e n t o f f r i c t i o n i n m o s t r o c k s h a s b e e n e x p e r i m e n t a l l y d e t e r m i n e d t o

ra n g e f ro m 0 .5 to 1 .0 (B y e r l e e , 1 9 6 8 ; H a n d in , 1 9 6 9 ; J a e g e r a n d C o o k , 1 9 7 9 , p . 5 9 ; B ra c e a n d

K o h l s t e d t , 1 9 8 0 ) , 2 w i l l b e l e s s t h a n 4 5 ° a n d th e a n g l e b e tw e e n c o n ju g a t e f a u l t s w i l l b e l e s s

1 3t h a n 9 0 ° w h e re b i s e c te d b y F a n d g re a t e r th a n 9 0 ° w h e re b i s e c te d b y F .

A n d e r s o n r e a s o n e d th a t i n n a tu ra l f a u l t s o n e o f th e p r in c ip a l s t r e s s o r i e n t a t i o n s w i l l b e

v e r t ic a l s i n c e t h e s u r fa c e o f t h e e a r t h m a y b e t h o u g h t o f a s a f re e s u r f a c e u n a b l e t o s u p p o r t

s h e a r s t r e s se s . T h e o th e r tw o p r in c ip a l s t r e s se s a re t h u s r e q u i r e d to b e h o r i z o n t a l . A s su m in g

t h a t th e r e l a t i v e m a g n i t u d e s o f t h e p r i n c i p a l s t r e s s e s m u s t c h a n g e f o r f a u l t in g t o o c c u r , t h e r e

a r e t h r e e p o s s i b l e r e l a t i o n s h i p s b e t w e e n t h e m a g n i t u d e s o f th e h o r i z o n t a l p r i n c i p a l s t re s s e s - -

t h e y a r e e i th e r b o th in c re a s in g in m a g n i tu d e , b o th d e c re a s in g in m a g n i tu d e , o r o n e i s i n c re a s in g

w h i l e t h e o th e r i s d e c re a s in g (A n d e r s o n , 1 9 5 1 , p . 1 3 ) . T h e s e c o r re s p o n d , r e s p e c t iv e ly , t o th e

t h r e e c o m m o n t y p e s o f c o n j u g a t e f a u l t s y s t e m s - - r e v e r s e f a u l t s , n o r m a l

fa u l t s , a n d w re n c h f a u l t s ( f i g u re 2 -2 ) . T h e s u c c e s s o f A n d e r s o n 's m e th o d i s w i tn e s se d b y th e

fa c t t h a t i t i s s t i l l u s e d to d a y a s a f i r s t a p p r o x i m a t io n fo r d e t e r m in i n g t h e p r in c ip a l s t r e s s

o r i e n t a t i o n s f ro m c o n ju g a t e f a u l t s e t s (D a v i s , 1 9 8 4 , p . 3 0 6 ; R a g a n , 1 9 8 5 , p . 1 3 5 ; S u p p e , 1 9 8 5 ,

p . 2 9 2 ; R o w la n d , 1 9 8 6 , p . 1 3 4 ; D e n n i s , 1 9 8 7 , p . 2 3 6 ; M a r s h a k a n d M i t r a , 1 9 8 8 , p . 2 6 1 ;

S p e n c e r , 1 9 8 8 , p . 1 9 9 ) . C a re m u s t b e u s e d , h o w e v e r , s in c e c o n ju g a t e f a u l t s e t s e x i s t w h ic h d o

n o t f i t t h e A n d e r s o n ia n c l a s s i f i c a t i o n ( O e r t e l , 1 9 6 5 ; A y d in , 1 9 7 7 ; A y d in a n d R e c h e s , 1 9 8 2 ;

R e c h e s a n d D i e t e r i c h , 1 9 8 3 ; K r a n t z , 1 9 8 8 ; K r a n t z , 1 9 8 9 ) .

2 .2 B o tt ' s F o r m u la

T h e n e x t im p o r ta n t s te p s i n p a l e o s t r e s s a n a l y s i s w e re t h e d e t e rm in a t i o n o f t h e

r e l a t i o n s h i p o f s h e a r s t r e s s e s t o t h e o r i e n t a t i o n o f f a u l t p l a n e s a n d t h e i r a s s o c i a t e d s l i p

d i r e c t io n s (W a l l a c e , 1 9 5 1 ) a n d th e r e la t i o n s h ip o f th e p r in c ip a l s t r e s s m a g n i tu d e s a n d

o r i e n t a t i o n s t o t h e r e s u l t in g d i r e c t i o n s o f m a x i m u m s h e a r s t re s s w i t h i n f a u l t p l a n e s ( B o t t ,

1 9 5 9 ) . T h e s e s t e p s w e re m o t iv a t e d b y th e f a c t t h a t ro c k in i t s n a tu ra l s t a t e i s r a r e ly in t a c t a n d

i s o t ro p ic (A n d e rs o n 's a s s u m p t io n ) . T h e u p p e r 5 t o 1 0 k i lo m e te r s o f th e e a r th 's c ru s t i s r i d d l e d

w i th p r e e x i s t i n g fa u l t p l a n e s , j o i n t s , a n d b e d d in g s u r fa c e s w i th s l id i n g o f t e n o c c u r r in g o n t h e s e

p la n a r d i s c o n t in u i t i e s lo n g b e fo re a s t a t e o f s t r e s s h ig h e n o u g h to c a u s e f r a c t u r e in a n in ta c t

v o lu m e o f ro c k i s r e a c h e d ( W a l l a c e , 1 9 5 1 ; B o t t , 1 9 5 9 ; J a e g e r , 1 9 6 0 ; D o n a th , 1 9 6 4 ; H a n d in ,

1 9 6 9 ; M c K e n z i e , 1 9 6 9 ) .

W a l l a c e (1 9 5 1 ) , u s in g s te r e o g ra p h i c p ro j e c t io n s , p lo t t e d s h e a r s t r e s s m a g n i tu d e s fo r

v a r io u s o r i e n t a t i o n s o f p l a n e s w i th in a s t r e s s s y s te m a n d th e d i r e c t i o n s o f m a x im u m s h e a r in g

s t r e s s in th o s e sa m e p l a n e s . W a l l a c e ' s m a jo r c o n t r ib u t io n , h o w e v e r , w a s to s h o w h o w a b o d y

h a s a t e n d e n c y t o s h e a r in a p la n e w h ic h r e p re s e n t s a c o m p ro m is e b e t w e e n e x p e r i e n c i n g a lo w

n o rm a l s t r e s s a n d a h ig h s h e a r s t r e s s a n d th a t t h i s p l a n e w i l l a lw a y s b e o r i e n t e d a t l e s s t h a n 4 5 °

1f ro m th e F d i r e c t i o n . T h i s m a y b e s h o w n b y s o l v i n g f o r t h e m i n i m u m o f t h e n o r m a l

F ig u r e 2 - 2 - T h e th re e A n d e rs o n ia n c la s s e s o f c o n ju g a te fa u l t s e t s . A . C o n j u g a t e t h r u s t f a u l t s

1 3 1w i t h h o r i z o n t a l F a n d v e r t ic a l F . B . C o n j u g a t e n o r m a l fa u l t s w i t h v e r t ic a l F a n d h o r i z o n ta l

3 1 3 2F . C . C o n ju g a t e w re n c h f a u l t s w i th F a n d F b o t h h o r i z o n t a l . In a l l t h r e e c a s e s F p a r a l l e l s

th e in te r s e c t i o n l i n e o f t h e tw o fa u l t p la n e s .

s t r e s s / s h e a r s t r e s s d i f f e re n c e in th e fo l l o w in g w a y

n sd [F - F ] / d2 = 0 (1 3 )

n sw h i c h , u p o n s u b s t i t u t i o n o f t h e s t a n d a r d f o r m u l a s f o r F a n d F (M e a n s , 1 9 7 6 , p . 7 2 ) b e c o m e s

1 3 1 3 1 3d [ ( (F + F ) / 2 ) + ( (F - F ) / 2 ) c o s ( 22) - ( (F - F ) / 2 ) s i n ( 22) ] / d2 = 0 (1 4 )

w h ic h , a f t e r d i f f e r e n t i a t i o n , y i e ld s

1 3(F - F ) [ - s i n ( 22) - c o s ( 22) ] = 0 (1 5 )

1 3w h i c h i m p l i e s t a n ( 22) = - 1 o r 2 = 6 7 .5 f o r a n y v a l u e s o f F a n d F ( f i g u r e 2 - 3 ) .

W a l l a c e 's w o rk l a i d t h e g ro u n d w o rk fo r B o t t (1 9 5 9 ) w h o d e r iv e d a fo rm u la r e l a t i n g th e

d i r e c t i o n o f m a x i m u m s h e a r s t r e s s w i t h i n t h e f a u l t p l a n e t o t h e f a u l t p l a n e 's o r i e n t a t i o n w i t h

re s p e c t to th e p r in c ip a l s t r e s s a x e s a n d th e r e l a t i v e m a g n i tu d e s o f t h e s e s t r e s se s w h ic h m a y b e

re p re s e n te d a s fo l l o w s

1 2 2 2 1 32 = t a n [ ( l l - M l + M l ) / ( l l ) ] (1 6 )-1 2 3

w h e re 2 i s t h e p i t c h a n g l e b e tw e e n t h e m a x im u m s h e a r s t r e s s d i r e c t i o n a n d th e s t r i k e o f th e

1 2 3f a u l t p la n e , l , l , a n d l a re th e t h re e d i r e c t i o n c o s in e s o f th e n o rm a l v e c to r to th e f a u l t p la n e ,

a n d M i s d e f i n e d a s ( A n g e l i e r , 1 9 7 9 ; M i c h a e l , 1 9 8 4 )

2 3 1 3M = [ (F - F ) / (F - F ) ] (1 7 )

w h ic h r a n g e s f ro m 0 .0 to 0 .1 a n d re p re s e n ts th e r e l a t i v e m a g n i tu d e s o f t h e th re e p r in c ip a l

s t r e s s e s ( i . e . d e s c r ib e s th e s h a p e o f th e s t r e s s e l l i p s o i d ) . O th e r s im i l a r p r in c ip a l s t r e s s

m a g n i tu d e r a t i o s h a v e b e e n d e f in e d , i n c lu d in g th e t e n s o r a s p e c t r a t i o * (C é l é r i e r , 1 9 8 8 )

1 2 1 3* = [ (F - F ) / (F - F ) ] (1 8 )

w h e re * a l s o r a n g e s f ro m 1 .0 t o 0 .0 ( i . e . * = 1 - M ) a n d th e p a ra m e te r R (c a l l e d C b y

A le k s a n d ro w s k i , 1 9 8 5 ) , w h ic h h a s b e e n d e f i n e d (A rm i j o a n d C is te rn a s , 1 9 7 8 ; E tc h e c o p a r , e t .

a l . , 1 9 8 1 ) a s

z x y xR = [ (F - F ) / (F - F ) ] (1 9 )

x y z 1 2 3R m a y h a v e v a lu e s f ro m -4 t o +4 s i n c e F , F , a n d F m a y c o r re s p o n d to F , F , a n d F

i n a n y o r d e r a l t h o u g h s o m e c o n f u s i o n h a s r e s u l t e d d u e t o t h e u s e o f th e s y m b o l R w i t h

d e f in i t i o n s d i f f e r e n t th a n th e o r ig in a l o n e o f A rm i jo a n d C i s te rn a s (L i s le , 1 9 8 7 ; G e p h a r t a n d

F o r s t y t h , 1 9 8 4 ; L a r ro q u e a n d L a u r e n t , 1 9 8 8 ) . A n i m p o r t a n t im p l i c a t i o n o f B o t t 's f o r m u l a i s

t h a t t h e s l i p d i re c t i o n o f a fa u l t p la n e i s d e p e n d e n t u p o n th e r e l a t i v e m a g n i tu d e s o f th e t h re e

1 2 3p r in c ip a l s t r e s s e s F , F , a n d F ( e x p re s se d b y M ) a n d n o t s im p ly th e i r o r i e n ta t i o n s . A

c o m p l e t e d e r i v a t i o n o f B o t t 's f o r m u l a a n d i t s a p p l i c a t i o n i n g e n e r a t i n g a r t i f i c i a l fa u l t

p o p u la t i o n s i s g iv e n in c h a p t e r 4 .

2 .3 G r a p h ica l M e th o d s o f F a u l t -S tr ia t io n P a leo st r e ss A n a ly s i s

O n e o f th e s im p le s t g r a p h ic a l m e t h o d s o f p a l e o s t r e s s a n a l y s i s u s i n g f a u l t - s t r ia t io n d a ta ,

i s t o p l o t t h e f a u l t p l a n e s o n a S c h m i d t o r W ü l f f s t e r e o g r a p h i c p r o j e c t i o n a l o n g w i t h t h e i r

F ig u r e 2 - 3 - A p lo t o f th e s h e a r s t r e s se s a n d n o rm a l s t r e s se s a c t i n g u p o n a p la n e p a r a l l e l t o th e

2 1F d i r e c t i o n v e r s u s th e i n c l in a t io n a n g le o f th e p la n e 's n o rm a l f r o m th e F o r i e n t a t i o n . T h e

u p p e r c u rv e re p re s e n t s th e n o r m a l s t r e s s e s a n d th e lo w e r c u r v e r e p re s e n t s t h e s h e a r s t r e s s e s .

I t m a y b e s e e n b y in sp e c t io n th a t t h e n o rm a l s t r e s s / s h e a r s t r e s s d i f f e re n c e i s a t a m in im u m a t

6 7 .5 ° (d e n o te d b y th e d a s h e d l i n e ) .

1a s s o c ia t e d s l i p d i r e c t i o n s . I f t h e f a u l t p o p u la t i o n fo rm s a c o n ju g a te s e t , t h e F a x i s i s a s s u m e d

3 2to b i s e c t t h e a c u te a n g le o f t h e c o n ju g a te s e t , F t h e o b tu s e a n g le , a n d F i s l o c a t e d a t t h e

in t e r s e c t i o n o f t h e fa u l t p la n e s - - a s s u m in g , o f c o u rs e , th a t t h e s e p la c e m e n ts a re c o n s i s te n t w i th

th e s l i p d i r e c t i o n s o n th e f a u l t p la n e s p r e s e n t ( f i g u re 2 -4 ) . T h e d ra w b a c k o f th i s m e th o d i s t h a t

i t w i l l o n ly w o rk o n th e s im p le s t o f c o n ju g a t e f a u l t s e t s (R a g a n , 1 9 8 5 , p . 1 3 5 ; S u p p e , 1 9 8 5 , p .

2 9 2 ; R o w l a n d , 1 9 8 6 , p . 1 3 4 ; M a r s h a k a n d M i t ra , 1 9 8 8 , p . 2 6 1 ) .

A s o m e w h a t d i f f e re n t t y p e o f g ra p h ic a l p a le o s t r e s s a n a ly s i s w a s d e v e lo p e d f r o m a

p o s tu l a t e d d i r e c t r e l a t io n s h i p b e tw e e n t h e r e g i o n a l s t r a i n e l l i p s o i d a n d th e r e g io n a l s t r e s s

e l l i p s o id a s so c i a t e d w i th a f a u l t p o p u la t i o n . A r th a u d 's m e th o d (A r th a u d , 1 9 6 9 ) , a n d th e

m o d i f i c a t i o n o f th a t m e th o d b y A le k s a n d ro w s k i (A le k s a n d r o w s k i , 1 9 8 5 ) , u s e d m o v e m e n t

1 2 3p l a n e s t o d e t e r m i n e t h e o r i e n t a t i o n s o f F , F , a n d F .

A n M -p la n e , o r m o v e m e n t p l a n e , a s so c i a t e d w i th a f a u l t i s t h e p l a n e c o n t a in in g th e

fa u l t ' s n o rm a l a n d s l i p v e c to rs ( f i g u re 2 -5 ) . O n e o f t h e im p o r t a n t p ro p e r t i e s o f m -p la n e s i s t h a t

t h e s e p la n e s c o n ta in a t l e a s t o n e o f t h e p r in c ip a l s t r a i n a x e s . A s su m in g a f a u l t p o p u la t i o n

c o n s i s t i n g o f r a n d o m ly -d i s t r i b u t e d , p re e x i s t i n g p la n e s a c t i v a t e d d u r in g o n e t e c t o n ic e v e n t , t h e

fo l lo w in g s t e p s a l l o w o n e to u s e m -p la n e s to g ra p h ic a l ly lo c a t e th e p r in c ip a l s t r a in a x e s

( A r t h a u d , 1 9 6 9 ; A l e k s a n d r o w s k i , 1 9 8 5 ) :

1 . P l o t t h e f a u l t p l a n e n o r m a l a n d s l i p v e c t o r s o n a s t e r e o n e t .

2 . J o i n e a c h p o l e a n d i t s a s s o c i a t e d s l i p v e c t o r w i t h a g r e a t c i r c l e . T h e s e g r e a t c i r c l e s a r e

th e m -p l a n e s .

3 . P lo t th e p o l e s (BM -p o le s ) to t h e m -p l a n e s .

A l l o f th e m -p la n e s s h o u ld in te r s e c t a t o n e , tw o , o r th re e g e n e ra l l y d i f fu se p o in t s w h ic h

a r e t h e n o r m a l s o f t h e s a m e n u m b e r o f m u t u a l l y p e r p e n d i c u l a r g r e a t c i r c l e s o f BM -

F ig u r e 2 - 4 - L o w e r -h e m i s p h e r e s t e r e o g r a p h i c p r o j e c t i o n s h o w i n g 2 4 c o n j u g a t e n o r m a l fa u l t s

a n d th e i r a s so c i a t e d s l i p v e c to r s ( s m a l l c i r c l e s o n f a u l t p l a n e s ) . In th i s s im p le f a u l t p o p u la t i o n ,

1 2 3 1t h e F , F , a n d F a x e s m a y b e a s s i g n e d b y i n s p e c t i o n - - F b i s e c t in g th e a c u te a n g l e o f th e

3 2c o n ju g a te fa u l t s e t , F b i s e c t in g th e o b tu s e a n g le , a n d F a t t h e in t e r s e c t io n o f t h e f a u l t p l a n e s .

F ig u r e 2 - 5 - L o w e r -h e m is p h e re s t e re o g ra p h ic p ro je c t io n s h o w in g th e re l a t i o n s h ip b e tw e e n

a f a u l t p l a n e , t h e n o rm a l a n d s l i p v e c to r s , a n d th e m -p la n e . T h e m -p la n e i s t h e p l a n e

p e rp e n d ic u la r to th e f a u l t a lo n g w h ic h m o v e m e n t t a k e s p la c e .

p o le s ( i . e . i f m - p l a n e s i n t e r s e c t , t h e i r BM -p o le s l i e o n a g re a t c i r c l e ) . In t e r s e c t io n p o in t s o f

th e m -p la n e s c o r re s p o n d to a n o r t h o g o n a l s y s te m o f X , Y , a n d Z a x e s w h e re X i s t h e a x i s o f

m a x im u m e x te n s io n , Y in th e i n t e rm e d ia te a x i s , a n d Z i s t h e a x i s o f m a x im u m s h o r t e n in g .

T h e s e a re t h e p r in c ip a l s t r a i n a x e s a n d a re a s s ig n e d a c c o r d in g to th e m o v e m e n t d i r e c t i o n s o f

t h e f a u l t s p r e s e n t a n d o r i e n t a t i o n o f a n y s t y l o l i t e s o r t e n s i o n f ra c t u r e s p r e s e n t ( f i g u r e 2 - 6 ) .

T h is t e c h n iq u e i s k n o w n a s A r th a u d 's m e th o d ( A r th a u d , 1 9 6 9 ) a n d h e c o n te n d e d th a t

g i v e n t h e s t r a i n e l l ip s o id a s s o c i a t e d w i t h a f a u l t p o p u l a t i o n , i t w a s p o s s i b l e t o p l a c e c o n s t r a i n t s

u p o n t h e s t r e s s e l l ip s o i d . A s e r i o u s l i m i t a t i o n i n t h i s m e t h o d i s th a t i t c a n b e s u c c e s s f u l l y

a p p l i e d o n l y t o p o p u la t io n s o r ig i n a t in g i n r a d i a l s t re s s f i e l d s ( i . e . o n e d e f in e d b y a M v a lu e o f

2 3 2 10 .0 w h e re F = F o r a M v a lu e o f 1 .0 w h e re F = F ) a n d th e o n ly a x i s o f d e fo rm a t io n

o b ta i n a b l e f ro m s u c h p o p u l a t io n s c o r re s p o n d s t o th e r e v o l u t io n a x i s o f a p ro la te o r o b l a t e s t re s s

e l l ip s o i d ( C a r e y , 1 9 7 6 ; A l e k s a n d r o w s k i , 1 9 8 5 ) .

A l e k s a n d r o w s k i (1 9 8 5 ) m o d i f ie d A r t h a u d 's m e t h o d t o m a k e i t a p p l i c a b l e f o r a g e n e r a l ,

t r i a x i a l s t r e s s f i e l d ( i . e . a s t r e s s f i e l d i n w h i c h t h e t h r e e p r i n c i p a l s t r e s s m a g n i t u d e s a r e

u n e q u a l ) . T h e p ro c e d u r e i s t h e s a m e a s in A r th a u d 's m e th o d ( s te p s 1 6 3 a b o v e ) e x c e p t th a t t h e

f in a l r e s u l t c o n s i s t s o f m o r e t h a n th r e e c o m m o n in t e r s e c t io n p o i n t s o f m -p l a n e s . E a c h o f th e s e

c o m m o n in t e r s e c t io n p o in t s m u s t t h e n b e s e p a ra t e ly a n a ly z e d t o a s c e r t a in w h e th e r o r n o t th e

s l i p v e c to rs c o r re s p o n d in g to t h e m -p la n e s l i e o n a g re a t c i r c l e a n d a t f a i r l y l a r g e a n g u la r

d i s ta n c e s f ro m o n e a n o th e r ; t h e i n t e r s e c t io n p o in t o f th i s g re a t c i r c l e w i th th e g i rd l e o f

a s so c i a t e d BM - p o l e s i s o n e o f t h e t h r e e p r i n c i p a l s t r e s s a x e s . T h e p l a n e p e r p e n d i c u l a r to t h i s

p r in c ip a l s t r e s s a x i s w h ic h p a s s e s th ro u g h th e c o m m o n in t e r s e c t io n po in t c o n ta in s th e o th e r tw o

p r i n c i p a l s t r e s s e s ( f i g u re 2 -7 ) . I f a n o th e r c o m m o n in te r s e c t io n p o in t c a n b e fo u n d w h ic h

s a t i s f i e s th e se c o n d i t i o n s , t h e t h re e p r in c ip a l s t r e s s a x e s m a y b e l o c a t e d w i th v a ry in g d e g re e s

o f p re c i s io n (A le k s a n d ro w s k i , 1 9 8 5 ; M a rs h a k a n d M i t r a , 1 9 8 8 , p . 2 6 3 ) . A M v a lu e m a y th e n

a l s o b e c a l c u l a t e d f ro m t h e o r i e n t a t i o n o f a n y o n e o f t h e s l i p v e c t o r s a n d i t s a s s o c i a t e d f a u l t

F ig u r e 2 - 6 - L o w e r h e m is p h e re s te r e o g ra p h i c p ro j e c t io n d e m o n s t r a t i n g th e m -p la n e m e th o d

o f lo c a t in g a p r in c ip a l s t r a i n a x i s . F o u r f a u l t p l a n e s a re u s e d , t h e i r p o l e s (BF 1 6 BF 4 ) a n d s l i p

v e c t o r s (S 1 6 S 4 ) a r e c o n n e c t e d b y g r e a t c i rc l e s (m -p l a n e s ) a n d th e p o l e s t o th e m -p l a n e s (BM 1

6 BM 4 ) a r e s h o w n t o fo r m a g i r d l e . T h e i n t e r s e c t i o n o f t h e m - p l a n e s ( w h i c h i s a l s o t h e p o l e

o f th e g i r d l e ) i s o n e o f t h e t h r e e p r i n c i p a l s t ra i n a x e s ( f i g u r e m o d i f ie d f ro m A r t h a u d , 1 9 6 9 ) .

F ig u r e 2 - 7 - A n a l y z i n g a c o m m o n i n t e r s e c t i o n p o i n t ( C I P ) o f m - p l a n e s ( s o l i d g r e a t c i r c l e s ) .

T h e p o l e s (BM -p o le s ) t o t h e m -p la n e s (c i r c l e s ) d e f in e a g re a t c i r c l e d e n o t e d b y G C P ( lo n g

d a s h e d / s h o r t d a s h e d l i n e ) a n d th e p o l e o f t h e G C P i s t h e C IP . A g re a t c i r c l e m a y b e d r a w n

t h r o u g h th e s l i p v e c t o r s (x s y m b o l s ) o f e a c h o f t h e m -p la n e s a n d i s d e n o t e d b y G C F ( s h o r t

d a s h e d l in e ) . T h e in t e r s e c t io n o f t h e G C P a n d G C F i s a p r in c ip a l s t r e s s a x i s F . T h e p l a n e

p e rp e n d i c u l a r to F , t h ro u g h th e C IP , a n d d e n o te d b y FP ( lo n g d a sh e d l i n e ) c o n ta in s th e o th e r

tw o p r i n c ip a l s t r e s s a x e s ( f i g u re m o d i f ie d f ro m A le k s a n d ro w s k i , 1 9 8 5 ) .

p l a n e u s i n g B o t t 's f o r m u l a ( e q u a t i o n 1 5 ) .

2 .4 R ig h t - D ih e d r a M e th o d s o f F a u l t -S tr ia t io n A n a ly s i s

A n o th e r g ra p h i c a l m e th o d o f p a l e o s t r e s s a n a ly s i s h a s b e e n d e v e lo p e d b y a d a p t in g th e

c o n s t ru c t io n t e c h n i q u e s o f f a u l t -p l a n e s o lu t io n s f ro m s e i s m ic d a t a t o s t r i a t e d f a u l t p o p u la t i o n s

( M c K e n z i e , 1 9 6 9 ; A n g e l i e r a n d M e c h l e r , 1 9 7 7 ; L i s l e , 1 9 8 7 ; L i s l e , 1 9 8 8 ) . T h e r e l a t i o n s h i p

b e t w e e n t h e f a u l t p l a n e s o l u t i o n s a n d t h e p r i n c i p a l s t r e s s a x e s w a s f i r s t s h o w n b y M c K e n z i e

1(1 9 6 9 ) , w h o r ig o ro u s ly d e m o n s t r a t e d t h a t th e m o s t c o m p r e s s iv e p r in c ip a l s t r e s s F m u s t l i e

w i th in th e q u a d ra n t c o n ta in in g t h e a x i s o f c o m p re s s io n P fo r f a u l t p l a n e s o lu t io n s o f s h a l lo w

e a r t h q u a k e s a s s u m e d to h a v e o c c u r r e d a lo n g p re e x is t i n g p la n a r d i s c o n t i n u i t i e s .

F a u l t -p l a n e s o l u t io n s a r e c o n s t ru c t e d f ro m s t r i a t e d fa u l t s o r f ro m th e s e i s m ic f i r s t

m o t io n s o f e a r th q u a k e s a n d sh o w th e r e l a t i o n s h ip b e tw e e n a f a u l t a n d i t s c o r re s p o n d in g

a u x i l i a ry p la n e , t h e z o n e s o f c o m p re s s io n a n d d i l a t i o n , a n d th e a x e s o f c o m p re s s io n a n d te n s io n

( f ig u re 2 -8 ) . T h e a u x i l i a ry p l a n e i s t h e p l a n e w h ic h i s p e rp e n d i c u l a r to b o th th e f a u l t p l a n e a n d

t h e s l i p d i r e c t i o n a n d , a l o n g w i th t h e f a u l t p l a n e , d e f in e s tw o c o m p re s s io n a l a n d t w o

e x t e n s io n a l r i g h t d ih e d ra (C o x a n d H a r t , 1 9 8 6 , p . 1 9 7 ) . B y c o n s t ru c t in g fa u l t p l a n e s o lu t io n s

fo r e a c h d a t u m o f a p o p u la t i o n o f f a u l t s , t h e o v e r l a p p in g q u a d ra n t s c o n t a in in g th e c o m p re s s io n

1a x i s P w i l l a c t to c o n s t r a i n t h e l o c a t i o n o f F . T h i s t e c h n iq u e h a s b e e n c a l l e d

th e r i g h t -d ih e d ra m e th o d ( la m é th o d e d e s d ièd re s d ro i t s ) b y A n g e l i e r ( A n g e l i e r a n d M e c h l e r ,

1 9 7 7 ) a n d w a s l a t e r m o d i f i e d b y L i s l e (L i s l e , 1 9 8 7 ; L i s l e , 1 9 8 8 ) w h o a d d e d a n a d d i t i o n a l

1c o n s t r a i n t u p o n t h e l o c a t i o n o f F .

T o u t i l i z e t h e s e m e t h o d s , c o n s i d e r a f a u l t p l a n e w i t h a n o r m a l v e c t o r N , a s l i p v e c to r

S , a n d a v e c t o r O a t r i g h t a n g le s to b o th N a n d S . T h re e o r th o g o n a l p l a n e s m a y b e d e f in e d b y

th e s e v e c t o r s - - t h e f a u l t p l a n e w h i c h c o n t a i n s S a n d O , t h e a u x i l i a ry p la n e w h ic h c o n t a in s

F ig u r e 2 - 8 - S c h e m a t i c d i a g ra m a n d lo w e r -h e m is p h e re s te r e o g ra p h i c p ro j e c t io n i l l u s t r a t i n g

th e r e l a t i o n s h ip b e tw e e n a n o rm a l f a u l t p la n e d ip p in g a t 4 5 ° a n d i t s a s s o c ia t e d a u x i l i a ry p l a n e ,

t h e z o n e s o f c o m p re s s io n a n d d i l a t i o n , a n d th e a x e s o f c o m p re s s io n a n d t e n s io n in a f a u l t p l a n e

s o lu t i o n .

N a n d O , a n d t h e m o v e m e n t p l a n e ( t h e m -p l a n e o f A le k s a n d r o w s k i , 1 9 8 5 ) w h i c h c o n t a i n s N a n d

S ( f i g u re 2 -9 ) . T h e fo u r r i g h t d ih e d ra o f A n g e l i e r a n d M e c h le r ' s m e th o d (A n g e l i e r a n d

M e c h l e r , 1 9 7 7 ) a re b o u n d e d b y t h e f a u l t p l a n e a n d t h e a u x i l i a ry p la n e . B y k n o w in g th e

d i r e c t io n , a n d s e n se , o f s l i p o n th e f a u l t p la n e , tw o d ih e d ra m a y b e d e f in e d a s c o m p re s s io n a l

r e g io n s a n d tw o d ih e d ra a s e x te n s io n a l r e g io n s . I f t h e a s su m p t io n i s m a d e th a t t h e

1 3c o m p re s s io n a l r e g io n s c o n t a in F a n d th a t th e e x te n s io n a l re g io n s c o n t a in F (M c K e n z i e ,

1 9 6 9 ) , t h e p o s i t i o n o f th e p a l e o s t r e s s a x e s m a y th u s b e c o n s t r a i n e d f o r e a c h fa u l t d a tu m .

1 3 1 3S u p e r im p o s in g th e F a n d F r e g i o n s f o r s e v e r a l f a u l t s , t h e p o s s i b l e p o s i t i o n s f o r F a n d F m a y

b e c o n s t r a in e d e v e n fu r t h e r ( f ig u re 2 -1 0 ) .

1 3L i s l e ( 1 9 8 7 ) , i n t r o d u c e d a n o t h e r c o n s t r a i n t u p o n t h e o r i e n t a t i o n s o f F a n d F b y

c o n s i d e r i n g h o w t h e o r i e n t a t i o n o f th e s l i p v e c t o r S c h a n g e s a s th e s t r e s s r a t i o M ( e q u a t io n 1 7 )

c h a n g e s . I f th e n o r m a l v e c t o r N o f a f a u l t p l a n e h a s d i r e c t i o n c o s in e s o f l , m , a n d n w i th re s p e c t

1 2 3t o th e F , F , a n d F a x e s r e s p e c t i v e l y , i t c a n b e s h o w n ( J a e g e r , 1 9 6 9 , p . 1 8 ) t h a t th e v e c t o r O

h a s d i r e c t i o n c o s in e s p ro p o r t i o n a l to

3 2m n (F - F ) ,

1 3n l (F - F ) , a n d (2 0 )

2 1lm (F - F ) .

2W h e n th e s t r e s s r a t i o M i s e q u a l to 0 .0 ( t h e c a s e o f a x ia l c o m p re s s io n ) , F i s e q u a l to

3F a n d th e f i r s t d i r e c t i o n c o s in e in e q u a t io n (2 0 ) r e d u c e s t o z e r o . T h i s im p l i e s th a t t h e v e c t o r

1 1O h a s n o c o m p o n e n t p a ra l l e l t o th e F a x i s , a n d F i s t h u s p a ra l l e l t o th e m o v e m e n t p l a n e . T h e

1p r o j e c t i o n o f F o n t o t h e f a u l t p l a n e w i l l t h e n b e c o i n c i d e n t w i t h t h e s l i p v e c t o r S . W h e n th e

2 1s t r e s s ra t i o i s e q u a l t o 1 .0 ( th e c a s e o f a x ia l e x te n s io n ) , F i s e q u a l to F a n d th e t h i rd

F ig u r e 2 - 9 - L o w e r -h e m is p h e re s t e re o g ra p h ic p ro je c t io n s h o w in g th e re l a t i o n s h ip b e tw e e n

a f a u l t p l a n e w i t h a n o r m a l v e c t o r N , i t s a s s o c i a t e d a u x i l i a r y p l a n e w i t h a n o r m a l v e c t o r S ( t h e

f a u l t 's s l i p v e c t o r ) , a n d i t s a s s o c i a t e d m o v e m e n t p l a n e ( m - p l a n e ) w i t h a n o r m a l v e c t o r O .

F ig u r e 2 -1 0 - L o w e r -h e m is p h e re s te re og ra ph ic p ro je c t io n s s h o w in g h o w th e r e g io n c o n ta in in g

1F i s c o n s t r a i n e d b y n o r m a l fa u l t p o p u l a t i o n s c o n s i s t i n g o f A . o n e p la n e , B . tw o p la n e s , a n d

C . th re e p la n e s .

d i r e c t i o n c o s i n e i n e q u a t i o n ( 2 0 ) re d u c e s t o z e r o . T h i s i m p l i e s t h a t th e v e c t o r O h a s n o

3 3c o m p o n e n t p a ra l l e l t o th e F d i r e c t i o n , a n d F i s t h u s p a ra l l e l t o th e m o v e m e n t p l a n e . T h e

3p r o j e c t i o n o f F o n t o t h e f a u l t p l a n e w i l l t h e n b e c o i n c i d e n t w i t h t h e s l i p v e c t o r S . S w i l l b e

w i t h i n t h e a c u t e a n g l e b e t w e e n t h e s e t w o e x t r e m e p o s i t i o n s f o r i n t e r m e d i a t e v a l u e s o f M

( f i g u r e 2 - 1 1 ) . G i v e n t h i s c o n s t r a i n t u p o n t h e p o s i t i o n o f S , i t c a n b e in fe r r e d th a t t h e

1 3p r o j e c t i o n s o f F a n d F l i e o n o p p o s i t e s i d e s o f S a n d a re b o th 9 0 ° o r l e s s f ro m S . O r , i n th re e

1 3d im e n s io n s , F a n d F w i l l b e w i th in s e p a r a t e r i g h t d ih e d ra b o u n d e d b y th e f a u l t ' s a u x i l i a ry a n d

m o v e m e n t p la n e s . L i s l e (1 9 8 7 ) a rb i t r a r i l y l a b e l l e d t h e se d i h e d ra A a n d B fo r c o n v e n ie n c e

1 3( f i g u r e 2 - 1 2 ) a n d , i f F i s k n o w n to l i e i n th e A d ih e d ro n , th e n F m u s t l i e i n t h e B a n d v i c e

v e rs a .

T o u s e L i s le ' s m e th o d , a u x i l i a r y a n d m o v e m e n t p l a n e s a re u s e d to c re a t e A a n d B

d i h e d ra fo r e a c h f a u l t . S u p e r im p o s in g th e s e d ih e d ra , r e g io n s a r e o b ta in e d w h ic h m a y b e

l a b e l l e d a c c o r d in g to w h ic h d ih e d ra i t f a l l s i n to fo r e a c h f a u l t . T h u s , i n a p o p u la t i o n o f fo u r

s u p e r i m p o s e d f a u l t s , t h e r e g i o n d e s i g n a t e d A B A A l ie s i n t h e A d i h e d r o n w i t h r e s p e c t to f a u l t

1 , i n th e B d ih e d ro n w i th r e s p e c t t o f a u l t 2 , a n d i n t h e A d i h e d ra w i th r e sp e c t t o f a u l t s 3 a n d 4 .

A s a n e x a m p le , c o n s id e r t h e d a t a s e t o f t h re e f a u l t s r e p re s e n te d in f i g u re 2 -1 3 . U s in g

1 3A n g e l i e r a n d M e c h le r ' s (1 9 7 7 ) m e th o d , f a i r l y l a rg e F a n d F r e g io n s m a y b e c o n s t ru c te d

( f ig u re 2 -1 4 ) . L i s le ' s (1 9 8 7 ) m e th o d b e g in s b y s u p e r im p o s in g th e A a n d B d ih e d ra ( f i g u re 2 -1 5 )

1 3a n d c o m p a r in g t h e m t o t h e F a n d F r e g i o n s o f A n g e l i e r a n d M e c h l e r 's ( 1 9 7 7 ) m e t h o d - -

1 3k e e p i n g i n m i n d t h a t F a n d F m u s t b e i n s e p a ra t e A a n d B d ih e d ra fo r e a c h fa u l t d a tu m . T h e

1F r e g io n in f i g u re 2 -1 4 c o n s i s t s o f th e A A A , A B A , A B B , B A A , B A B , B B A , a n d B B B a re a s o f

1 3f i g u r e 2 - 1 5 . T h e p r e s e n c e o f F i n a r e a B A B i s c o m p a t ib l e w i th F b e in g in A B A . T h e s a m e

1m a y b e s a id fo r A A A , A B A , B A B , B B A , a n d B B B in th e F r e g io n a n d A A A , A A B ,

F ig u r e 2 -1 1 - L o w e r -h e m i s p h e r e s t e r e o g r a p h i c p r o j e c t i o n s h o w i n g 2 0 s l i p v e c t o r s

re p re s e n t in g M v a lu e s r a n g in g f ro m 0 .0 to 1 .0 o n a f a u l t p la n e w i th a n o rm a l v e c to r o r i e n te d

1 2 3a t 7 0 /0 3 0 d e g re e s . T h e F , F , a n d F a x e s c o r re s p o n d to th e n o r th , u p , a n d e a s t d i r e c t i o n s

1 3r e s p e c t i v e l y . T h e d a s h e d l i n e s r e p r e s e n t th e p r o j e c t i o n s o f F a n d F o n to th e f a u l t p l a n e

( f ig u re m o d i f ie d f ro m S c h im m ri c h , 1 9 9 0 ) .

F ig u r e 2 -1 2 - L o w e r -h e m is p h e re s t e re o g ra p h ic p ro je c t io n s h o w in g th e r i g h t d ih e d ra d e f in e d

b y L i s l e (1 9 8 7 ) a s A ( s t i p p l e d ) a n d B (u n s t ip p l e d ) . T h e s e r i g h t d ih e d ra a re b o u n d e d b y t h e

fa u l t ' s a u x i l i a r y a n d m o v e m e n t p l a n e s . T h e p o l e t o th e f a u l t p l a n e i s N , t h e p o le t o th e

m o v e m e n t p l a n e i s O , a n d t h e p o l e t o th e a u x i l i a r y p l a n e i s S ( t h e fa u l t ' s s l i p v e c to r ) .

F ig u r e 2 -1 3 - L o w e r -h e m is p h e re s te r e o g r a p h ic p ro j e c t io n o f t h r e e f a u l t s w i th s l ip v e c t o r s S

1 3a n d f a u l t n o r m a l s N . T h e F /F a n d A / B d i h e d r a a r e d e f i n e d f o r e a c h f a u l t d a t u m ( f i g u r e

m o d i f ie d f ro m L is le , 1 9 8 7 ) .

1 3F ig u r e 2 -1 4 - L o w e r -h e m i s p h e r e s t e re o g ra p h ic p ro je c t io n s h o w in g th e F a n d F f i e l d s

c o n s t ru c te d b y s u p e r im p o s in g t h e f a u l t d a t a f ro m f ig u re 2 -1 3 . T h i s i s t h e A n g e l i e r a n d M e c h le r

(1 9 7 7 ) m e th o d ( f i g u re m o d i f ie d f ro m L is le , 1 9 8 7 ) .

F ig u r e 2 -1 5 - L o w e r -h e m is p h e re s t e re o g ra p h ic p ro je c t io n s h o w in g th e r e g io n s c o n s t ru c te d

b y s u p e r im p o s in g th e A a n d B d ih e d ra f ro m th e f a u l t d a t a i n f i g u re 2 -1 3 . T h i s i s L i s le ' s (1 9 8 7 )

c o n s t r a in t ( f i g u re m o d i f ie d f ro m L is le , 1 9 8 7 ) .

3 1A B A , B A B , a n d B B B in th e F r e g io n . T h e B A A a n d A B B a re a s in th e F r e g i o n , h o w e v e r ,

3h a v e n o c o u n te rp a r t s in th e F r e g i o n ( A B B o r B A A ) a n d m a y b e e l i m i n a t e d a s a r e a s o f p o s s i b l e

1 3F d i r e c t i o n s . F o r th e s a m e re a so n , B B A m a y b e e l im in a te d a s a n a re a c o n ta in in g th e F

1 3d i r e c t i o n . T h e r e s u l t in g s o l u t i o n ( f i g u r e 2 - 1 6 ) h a s n o t i c e a b l y s m a l l e r F a n d F r e g io n s .

I t i s e a s y to s e e t h a t fo r th e se m e th o d s to g iv e s a t i s f a c t o r y r e su l t s , o n e m u s t h a v e a

1 3p o p u la t i o n o f f a u l t s s u f f i c i e n t ly s c a t t e re d to c o n s t r a in th e F a n d F r e g io n s to s m a l l a re a s . A

p o p u l a t i o n o f f a u l t s w h e r e a l l o f th e f a u l t s h a v e s i m i l a r o r i e n t a t i o n s w i l l y i e l d n o m o r e

in fo rm a t io n th a n a n y s u b s e t o f th a t p o p u la t i o n ( f ig u re 2 -1 7 ) . T h e s e m e th o d s w i l l a l s o n o t

p e r fo rm v e ry w e l l w h e n d e a l in g w i th c e r t a i n ty p e s o f s y m m e t r i c a l f a u l t p o p u la t i o n s s u c h a s

c o n ju g a t e s e ts (A n d e r s o n , 1 9 5 1 ) o r o r th o rh o m b ic s e t s (A y d in a n d R e c h e s , 1 9 8 2 ; K ra n t z , 1 9 8 6 ;

K ra n tz , 1 9 8 9 ) o f fa u l t s ( f i g u re 2 -1 8 ) .

T h e g ra p h ic a l m e th o d s o f p a l e o s t r e s s a n a ly s i s d e v e lo p e d b y A n g e l i e r a n d M e c h le r

( 1 9 7 7 ) a n d L i s l e ( 1 9 8 7 ) a r e e x t r e m e l y c u m b e r s o m e t o d o o n a s t e r e o n e t w h e n d e a l i n g w i t h m o r e

th a n a h a n d fu l o f f a u l t p l a n e s . F o r th i s r e a so n , th e se m e th o d s a re u s u a l ly p e r fo rm e d

n u m e r i c a l ly b y a c o m p u te r p ro g ra m (A n g e l ie r a n d M e c h l e r , 1 9 7 7 ; L i s le , 1 9 8 8 ) . I t s h o u ld b e

k e p t in m i n d , h o w e v e r , t h a t e v e n t h o u g h t h e s e m e t h o d s a r e a d a p t e d f o r a c o m p u t e r , t h e y a r e

s t i l l e s se n t i a l l y c o n s id e re d g ra p h i c a l m e th o d s a n d n o t c o m p u ta t i o n a l m e th o d s .

1 3F i g u r e 2 - 1 6 - L o w e r -h e m is p h e re s t e re o g ra p h ic p ro je c t io n s h o w in g th e F a n d F f i e ld s

c o n s t r u c t e d b y u t i l i z i n g L i s l e ' s ( 1 9 8 7 ) c o n s t r a i n t o n t h e f a u l t d a t a f ro m f ig u r e 2 - 1 3 . E v e n w i t h

a s m a l l p o p u la t i o n o f t h re e f a u l t s , t h e re i s a n o t i c e a b le im p ro v e m e n t o v e r A n g e l i e r a n d

M e c h le r ' s (1 9 7 7 ) m e th o d ( f i g u re m o d i f ie d f ro m L is le , 1 9 8 7 ) .

1 3F ig u r e 2 -1 7 - L o w e r -h e m is p h e re s t e re o g ra p h ic p ro je c t io n s h o w i n g t h e F a n d F r e g io n s

d e r i v e d u s in g L is le 's (1 9 8 7 ) m e th o d o n fo u r t h ru s t f a u l t s a l l h a v in g a s im i l a r o r i e n ta t i o n .

1 3F ig u r e 2 -1 8 - L o w e r -h e m is p h e re s t e r e o g r a p h i c p ro je c t io n s h o w in g th e F a n d F r e g io n s

d e r iv e d u s in g L i s le ' s (1 9 8 7 ) m e th o d o n A . a c o n ju g a t e s e t o f fo u r n o rm a l f a u l t s a n d B . a n

o r t h o rh o m b ic s e t o f fo u r n o rm a l f a u l t s .

C H A P T E R 3

C O M P U T A T IO N A L P A L E O S T R E S S A N A L Y S IS O F F A U L T P O P U L A T IO N S

G iv e n th e a s s u m p t io n th a t f a u l t s w i l l s l i p i n t h e d i r e c t i o n o f th e i r m a x im u m re s o lv e d

s h e a r s t r e s s , d e te rm in in g th e d i r e c t i o n a n d s e n s e o f s l i p o n a p o p u la t i o n o f f a u l t p l a n e s o f

k n o w n o r i e n t a t i o n s f o r a g i v e n s t r e s s t e n s o r F i s a t r i v i a l m a t t e r (B o t t , 1 9 5 9 ) . T h e in v e r s e o f

t h i s p r o b l e m - - f in d i n g a s t r e s s t e n s o r F s a t i s fy in g k n o w n s l i p d i r e c t i o n s a n d o r i e n t a t i o n s fo r

a p o p u l a t i o n o f fa u l t s - - i s m u c h m o r e d i f f i c u l t . T h i s i s t e r m e d t h e " i n v e r s e p r o b l e m " a n d i t s

s o lu t i o n i s th e g o a l o f co m p u ta t io n a l m e th o d s o f p a le o s t r e s s a n a ly s i s (E tc h e c o p a r , e t . a l . , 1 9 8 1 ;

A m i j o , e t . a l . , 1 9 8 2 ; A n g e l i e r , 1 9 8 9 ) .

A l l p a le o s t r e s s a n a ly s i s m e th o d s a s su m e th a t tw o i t e m s o f in fo rm a t io n a re k n o w n fo r

e a c h fa u l t - - t h e fa u l t p la n e 's o r i e n ta t i o n in a g e o g ra p h ic c o o rd in a te s y s te m a n d th e fa u l t ' s

d i r e c t i o n a n d s e n s e o f s l i p . I n a d d i t i o n , tw o v e ry im p o r t a n t fu n d a m e n ta l a s su m p t io n s a r e m a d e

b y a l l o f th e m e th o d s - - t h a t t h e d i r e c t i o n o f s l i p o n a fa u l t p l a n e i s a lw a y s p a ra l l e l t o th e

d i r e c t io n o f r e s o lv e d s h e a r s t r e s s o n th a t p l a n e a n d t h a t a l l o f th e f a u l t s a r e a c t i v a t e d w i th in a

u n iq u e , s t a t i c s t r e s s f i e ld .

3 .1 E a r ly A tt e m p ts a t C o m p u ta t io n a l P a leo s t r e ss A n a ly s i s

In 1 9 7 4 , C a re y a n d B ru n ie r m a d e t h e f i r s t a t t e m p t a t fo rm u la t in g a n d s o lv in g th e

m a th e m a t i c s d e f i n in g th e in v e rs e p ro b le m (A rm i j o , e t . a l . , 1 9 8 2 ; C é l é r i e r , 1 9 8 8 ; A n g e l i e r ,

1 9 8 9 ) . T w o y e a r s l a t e r , C a re y (1 9 7 6 ) d e v e lo p e d th e f i r s t p a l e o s t r e s s a n a ly s i s p ro g ra m w h ic h

s o u g h t to m in im iz e t h e a n g u la r d e v ia t i o n s be tw e e n m e a su re d f a u l t s t r i a t i o n s a n d th e c a lc u la t e d

s h e a r s t re s s d i r e c t i o n s o n e a c h f a u l t p l a n e f o r a c h o s e n p a l e o s t r e s s t e n s o r F . A n g e l i e r a l so

d e v e lo p e d a s im i l a r m e th o d a t a p p ro x im a te ly th e s a m e t im e (1 9 7 5 ) . S in c e t h e n , A n g e l i e r h a s

d e v e l o p e d s e v e ra l s u c c e s s iv e m e th o d s , e a c h p o s se s s in g s l i g h t im p ro v e m e n t s i n th e

m a th e m a t i c a l a lg o r i th m s u s e d to p e r f o rm th e a n a ly s e s (c h a p te r 6 ) .

A n i m p o r t a n t c h a r a c t e r i s t i c ( s o m e w o u ld s a y a p r o b l e m ) o f m e t h o d s o f p a l e o s t r e s s

a n a l y s i s s u c h a s A n g e l i e r ' s i s t h a t t h e y s e t u p n o n - l i n e a r i t e r a t i v e e q u a t io n s a n d th u s h a v e

e x t r e m e ly c o m p le x m a th e m a t i c a l s o lu t io n s . T h e se e q u a t io n s a re t e rm e d n o n - l i n e a r b e c a u s e ,

fo r e a c h s te p i n t h e i t e r a t i o n , t h e o u tp u t v a r i a b l e i s c h a n g e d ( ju s t a s i t i s i n a l i n e a r e q u a t io n )

a n d t h i s n e w o u t p u t v a r i a b l e w i l l r e s u l t i n d i f fe r e n t i n p u t v a r i a b l e s ( w h i c h d o e s n o t h a p p e n i n

a l i n e a r e q u a t i o n ) .

3 .2 E tc h ec o p a r 's M e th o d o f P a leo st r e ss A n a ly s i s

In 1 9 8 1 , E tc h e c o p a r (E tc h e c o p a r , e t . a l . , 1 9 8 1 ) d e v e l o p e d a m e t h o d o f p a l e o s t r e s s

a n a l y s i s s i m i l a r t o t h a t b e i n g d e v e l o p e d a t a p p r o x i m a t e l y th e s a m e t i m e b y A n g e l i e r ( A n g e l i e r ,

1 9 7 9 ; A n g e l i e r , e t . a l . , 1 9 8 2 ) . T h i s m e th o d w a s s im i l a r to A n g e l i e r ' s i n th a t i t s o u g h t t o

m in im iz e t h e a n g u l a r d e v ia t i o n o f th e m a x im u m s h e a r s t r e s s d i r e c t i o n s f ro m th e s l i p d i r e c t i o n s

f o r a c h o s e n p a l e o s t r e s s t e n s o r F o n e a c h f a u l t p l a n e i n t h e p o p u l a t i o n e x a m i n e d . T h e o n l y

s u b s t a n t i a l d i f f e r e n c e w a s t h e u s e b y E tc h e c o p a r o f a s l i g h t ly d i f f e r e n t i t e r a t i v e a lg o r i t h m fo r

p e r f o rm in g th e n o n - l in e a r l e a s t - s q u a re s in v e rs io n .

A f t e r c o r re s p o n d in g w i t h A r n a u d E t c h e c o p a r o f th e U n iv e rs i t é d e s S c ie n c e s e t

T e c h n i q u e s d u L a n g u e d o c i n M o n tp e l l i e r , F r a n c e , I o b ta in e d a c o p y o f h i s p ro g ra m th o u g h

R ic h a rd P lu m b o f S c h lu m b e r g e r D o l l R e s e a rc h in R id g e f i e ld , C o n n e c t i c u t . T h e p ro g ra m

s o u rc e c o d e w a s w r i t t e n i n F O R T R A N a n d s e n t o n a m a g n e t i c t a p e , t h e c o n t e n t s o f w h ic h I

t r a n s fe r r e d to S U N Y A lb a n y 's V A X -8 6 5 0 m a in f ra m e c o m p u te r . A f t e r t r a n s l a t i n g t h e p ro g ra m

d o c u m e n t a t i o n f ro m t h e o r i g i n a l F r e n c h i n t o E n g l i s h ( w i t h t h e a s s i s t a n c e o f D e b r a L e n a r d - -

a S U N Y A l b a n y l in g u i s t i c s m a j o r ) , I w a s a b l e t o c o m p i l e a n d r u n t h e p r o g r a m f o r s e v e r a l f a u l t

p o p u la t i o n s . U n fo r tu n a te ly , a l l o f th e r e s u l t s o b t a in e d w e re n o d i f f e re n t th a n

th o s e g iv e n b y A n g e l i e r ' s p ro g ra m a n d I a b a n d o n e d fu r th e r t e s t i n g o f t h i s m e th o d in f a v o r o f

A n g e l i e r ' s .

3 .3 M ich a e l ' s M e th o d o f P a leo st r e ss A n a ly s i s

M ic h a e l (1 9 8 4 ) d e r iv e d a m e th o d o f p a l e o s t r e s s a n a ly s i s w h ic h m a d e a n e w in i t i a l

sa s s u m p t io n - - t h a t t h e m a g n i tu d e s o f th e s h e a r s t r e s s F o n e a c h o f th e f a u l t p l a n e s in a

p o p u la t i o n a t t h e t im e o f s l i p a r e s i m i l a r . M ic h a e l c l a im e d th i s a s su m p t io n w a s ju s t i f i e d b y

o b s e rv in g th a t t h e f a u l t p l a n e s a l l e x p e r i e n c e d s l i p , t h e r e fo re t h e a b s o lu t e m a g n i tu d e s o f th e

ss h e a r s t r e s se s o n a l l o f t h e p la n e s w e re s im i l a r a n d m in im iz in g th e d i f f e re n c e *F * - 1 f o r a l l

so f t h e f a u l t p la n e s w i l l a l l o w o n e to d e te rm in e *F * f o r e a c h f a u l t p l a n e . T h i s a l l o w s a f a i r l y

s i m p l e l in e a r in v e r s i o n t o s o l v e f o r t h e s t r e s s t e n s o r F .

I w ro t e a T u r b o P a s c a l v e r s io n 3 .0 1 c o m p u te r p ro g ra m fo r p e r fo r m in g p a l e o s t r e s s

a n a l y s e s u s in g M ic h a e l 's m e th o d in M a y , 1 9 8 7 . U n fo r tu n a t e ly , I fo u n d th a t t h i s m e th o d g a v e

in c o n s i s t e n t r e s u l t s fo r m a n y ty p e s o f f a u l t p o p u la t i o n s - - e s p e c i a l l y t h o s e w i th fa u l t s w h ic h

a re c lo s e t o b e in g p a ra l l e l t o t h e p r in c ip a l s t r e s s a x e s . T h i s i s p ro b a b ly d u e to th e f a c t t h a t t h e

s h e a r s t r e s s o n s u c h p la n e s b e c o m e s q u i t e l o w re l a t i v e t o t h o s e p la n e s a t 4 5 ° t o th e p r in c ip a l

s t r e s s a x e s ( f i g u re 3 -1 ) . M ic h a e l ' s m e th o d a l s o d o e s n o t w o rk w e l l w i th to o fe w fa u l t s (w h a t

c o n s t i t u t e s " to o fe w " i s n o t w e l l -d e f in e d a n d i s d e p e n d e n t u p o n th e f a u l t ' s o r i e n t a t i o n s ) a n d

fa u l t s w h ic h a l l h a v e a v e ry s im i l a r o r i e n ta t i o n . T h i s i s b e c a u s e th e i n v e rs io n m a t r ix b e c o m e s

c l o s e to b e in g a s i n g u l a r m a t r ix a n d th e c a l c u l a t e d c o n f id e n c e l i m i t s b e c o m e v e r y l a r g e a s a

re s u l t (M ic h a e l , 1 9 8 4 ) .

C é l é r i e r (1 9 8 8 ) s e v e re l y c r i t i c i z e d M ic h a e l ' s m e t h o d a n d s t a t e d th a t t h e s h e a r s t r e s s

a s su m p t io n d o e s n o t c o r re s p o n d to a r e a l i s t i c f a i l u re c r i t e r io n a n d th e o n ly ra t i o n a l fo r u s in g

i t i s t h a t i t r e s u l t s i n a s im p l i f i c a t i o n f o r t h e i n v e rs io n b y l i n e a r i z in g t h e e q u a t io n s . M ic h a e l

F ig u r e 3 - 1 - N o r t h e a s t q u a d ra n t o f a lo w e r -h e m is p h e re s te r e o g ra p h i c p ro j e c t io n s h o w in g

p o le s to f a u l t p l a n e s w h e re t h e r e l a t i v e s iz e s o f e a c h p o l e r e f l e c t t h e r e l a t i v e m a g n i tu d e s o f th e

s h e a r s t r e s s e s o n th e i r a s s o c i a t e d p l a n e s . A M v a l u e o f 0 .5 a n d a r a t io o f th e i s o t ro p i c s t re s s

to th e d e v ia to r i c s t r e s s o f 4 .5 (M ic h a e l , 1 9 8 4 ) w a s u s e d to c a lc u la te th i s d a ta .

la t e r m o d i f i e d h i s a l g o r i t h m to u s e n o n - G a u s s i a n r a th e r th a n G a u s s i a n s ta t i s t i c s in th e

in v e r s io n (M ic h a e l , 1 9 8 7 a ; M ic h a e l , 1 9 8 7 b ) w h i c h i m p r o v e d t h e m e t h o d b u t w h i c h a l s o

e f f e c t i v e ly re n d e re d m y p ro g ra m o b s o le te .

3 .4 G e p h a r t a n d F o r sy th 's M e th o d o f P a leo st r e ss A n a ly s i s

A ls o i n 1 9 8 4 , G e p h a r t a n d F o r s y th p r o p o s e d a s l ig h t ly d i f f e r e n t m e t h o d o f p a l e o s t r e s s

a n a l y s i s u s in g e a r th q u a k e fo c a l m e c h a n i sm d a t a . T h i s m e th o d b e g in s b y s e t t i n g u p a g r id o f

p o i n t s o n a s t e r e o n e t (G e p h a r t r e c o m m e n d s e i t h e r 1 0 ° o r 5 ° s p a c i n g s ) a n d t h e n s y s t e m a t i c a l l y

1o r ie n t in g th e m o s t c o m p r e s s iv e p r in c ip a l s t r e s s a x i s F s u c h th a t i t i s c o in c id e n t w i th e a c h o f

th e se g r id po in t s in tu rn w h i l e th e n s y s t e m a t i c a l l y o r i e n t in g th e l e a s t c o m p re s s iv e p r in c ip a l

3 1s t r e s s a x i s F s u c h th a t i t i s c o in c id e n t w i th a l l p o in t s 9 0 ° f ro m F i n t u rn a n d th e n v a ry in g th e

3s t r e s s r a t io M a t e a c h o f th e F l o c a t io n s . T h i s o b v io u s ly r e s u l t s i n a v e ry l a rg e n u m b e r o f

s t r e s s t e n s o rs b e in g d e f in e d fo r e a c h fa u l t p o p u la t i o n (d e p e n d e n t u p o n th e n u m b e r o f g r i d

p o i n t s d e f i n e d ) . F o r e a c h s t r e s s t e n s o r F t h u s d e f in e d , t h e a n g l e w h ic h th e n o rm a l v e c to r fo r

e a c h f a u l t p l a n e m u s t b e r o ta t e d t h ro u g h t o h a v e th a t f a u l t ' s s l i p v e c to r b e c o n s i s te n t w i th th e

s t r e s s f i e l d i s c o m p u te d . T h e p r o g r a m s e e k s t o m in i m iz e t h e s u m o f t h e s q u a re s o f th e s e

a n g u l a r d iv e rg e n c e s . T h e s t r e s s m o d e l w h ic h h a s th e s m a l l e s t s u m i s a s su m e d to b e th e

p o p u la t i o n 's p a le o s t r e s s t e n s o r .

I o b t a in e d a c o p y o f th i s p a l e o s t r e s s a n a ly s i s p ro g ra m in F e b r u a ry , 1 9 8 9 f ro m J o h n

G e p h a r t a t C o r n e l l U n i v e r s i ty . T h e p r o g r a m c o n s i s t e d o f s e v e r a l s u b r o u t i n e s a l l w r i t t e n i n

F O R T R A N f o r t h e M a c i n t o s h I I c o m p u t e r . A m a j o r c h a r a c t e r i s t i c o f t h e p r o g r a m i s th a t i t i s

v e ry c o m p u ta t io n a l ly in t e n s iv e a n d re q u i r e s a p p ro x im a te ly a fu l l 2 4 -h o u r d a y to p e r fo rm a

s e a r c h o f 1 0 ,0 0 0 s t r e s s m o d e l s f o r a r e l a t i v e l y s m a l l d a t a s e t o f 2 0 f o c a l m e c h a n i s m s . W h i l e

th i s i s n o t to o s e v e re a l im i t a t i o n fo r a sm a l l n u m b e r o f a n a ly se s , a th o ro u g h t e s t i n g o f th e

m e th o d w o u ld b e e x t r e m e ly t i m e c o n s u m in g .

G e p h a r t f u r th e r m o d i f i e d t h i s m e th o d a n d c re a t e d a F O R T R A N p ro g ra m c a l l e d F M S I

(a n a n c r o n y m fo r fo c a l m e c h a n i sm s t r e s s in v e r s io n ) w h ic h h a s re c e n t ly b e e n s u b m i t t e d fo r

p u b l i c a t i o n ( G e p h a r t , 1 9 9 0 ) . T h i s m e t h o d d o e s n o t d i f fe r m u c h f ro m e a r l ie r w o r k a n d i s

p r i m a r i ly a n a t t e m p t t o s p e e d u p th e c o m p u ta t i o n a l p ro c e d u re s .

3 .5 R e c e n t T r e n d s in P a leo st r e ss A n a ly s i s

In 1 9 8 7 , R e c h e s d e r iv e d a p a le o s t r e s s a n a ly s is m e th o d w h ic h u se d a l i n e a r l e a s t - s q u a re s

in v e r s io n m e th o d ra th e r th a n a n o n - l i n e a r o n e . T h i s g re a t ly s im p l i f i e s th e c a l c u l a t i o n s

in v o l v e d b y t a k in g th e C o u lo m b f a i l u r e c r i t e r io n i n t o a c c o u n t ( c h a p t e r 7 ) . C é l é r i e r (1 9 8 8 ) a l s o

d e r iv e d m a th e m a t i c a l a lg o r i t h m s fo r c a lc u la t in g pa le o s t r e s s t e n so rs b y in t ro d u c in g a f r i c t i o n a l

c o n s t r a in t u p o n th e f a u l t s i n th e p o p u la t i o n . T h e a d v a n ta g e o f b e in g a b l e to i n t r o d u c e a n

a d d i t i o n a l c o n s t r a in t i s t h a t t h e i n v e rs e p ro b le m e q u a t io n s b e c o m e l i n e a r a n d th u s m u c h e a s i e r

a n d fa s te r t o s o lv e c o m p u ta t i o n a l l y .

T h e m o s t r e c e n t p a p e r s o n p a l e o s t r e s s a n a l y s i s a r e c o n c e r n e d w i t h t e c h n i q u e s t o

1 2 3e s t im a te a b so lu te m a g n i tu d e s fo r t h e p r in c ip a l s t r e s s a x e s F , F , a n d F r a t h e r th a n s i m p l y

th e i r r e l a t i v e m a g n i tu d e s (e x p re s se d b y th e s t r e s s r a t io M ) . T o d o t h i s , tw o a d d i t i o n a l

c o n s t r a in t s m u s t b e p la c e d u p o n th e p ro b le m . A n g e l i e r (1 9 8 9 ) a t t e m p te d to d o th i s b y

c o n s id e r in g g e o lo g ic a l ly - r e a s o n a b le ru p tu re a n d f r i c t i o n l a w s fo r t h e f a u l t s e x a m in e d .

A n g e l i e r a l s o a t t e m p te d to c o n s t r a in th e v e r t i c a l s t r e s s b y a s su m in g i t t o b e c o in c id e n t w i th o n e

o f t h e t h r e e p r i n c i p a l s t r e s s e s a n d t a k i n g i n t o a c c o u n t t h e t h i c k n e s s o f t h e s e d i m e n t a r y

o v e rb u rd e n fo r f a u l t s i n th e B a s in a n d R a n g e H o o v e r D a m lo c a l i t y in N e v a d a - A r i z o n a

(A n g e l i e r , e t . a l . , 1 9 8 5 ; A n g e l i e r , 1 9 9 0 ) .

A n o th e r i m p o r t a n t a re a o f r e s e a rc h i s t h e a t t e m p t to r e l a t e f a u l t g e o m e t ry a n d

k in e m a t i c s to d r iv in g s t r e s se s . D y n a m ic ( s t r e s s -b a s e d ) a n d k in e m a t i c ( s t r a in -b a s e d ) m e th o d s

o f f a u l t a n a ly s i s a r e b e in g u s e d to g e th e r in a n a t t e m p t to l e a r n m o re a b o u t th e m e c h a n i c s o f

f a u l t in g . A c u r re n t l e a d e r in t h i s f i e l d o f s t u d y i s R i c h a r d A l l m e n d i n g e r o f C o r n e l l U n i v e r s i ty

(M a rr e t t a n d A l l m e n d in g e r , 1 9 9 0 ) .

C H A P T E R 4

P R O B L E M S IN P A L E O S T R E S S A N A L Y S IS

P r o b l e m s i n c o m p u t a t i o n a l m e t h o d s o f p a l e o s t r e s s a n a l y s i s u s i n g s t r ia t e d - fa u l t

p o p u la t i o n s a r i s e in tw o a re a s - - i n th e g a th e r in g o f d a t a a s in p u t fo r th e p ro g ra m s a n d in th e

fu n d a m e n ta l s im p l i fy in g a s s u m p t io n s m a d e b y th e se p ro g ra m s a s t h e y a t t e m p t t o c a lc u la t e a

p a le o s t r e s s t e n so r . A l l o f th e p a le o s t r e s s a n a ly s i s p ro g ra m s c u r re n t ly in u s e re q u i r e , a s

n u m e r i c a l i n p u t , t h e o r i e n ta t i o n o f e a c h f a u l t p la n e a lo n g w i th t h e f a u l t ' s d i r e c t i o n , a n d se n se ,

o f s l ip . P r o b l e m s m a y a r i s e i n g a t h e r i n g t h i s in f o r m a t i o n f ro m t h e f i e l d s i n c e i n a c c u r a t e d a t a

m a y r e s u l t i n a p r o g r a m r e t u r n i n g a n i n c o r re c t p a l e o s t r e s s t e n s o r ( s e c t i o n s 4 .1 a n d 4 . 2 ) .

P a l e o s t r e s s a n a ly s i s p ro g ra m s a l s o m a k e s e v e ra l s im p l i fy in g a s su m p t io n s a b o u t f a u l t s a n d th e

n a t u r e o f f a u l t in g w h i c h m a y l e a d t o r e s u l t a n t e r r o r s ( s e c t i o n s 4 . 3 t h r o u g h 4 . 6 ) . In o r d e r to

u n d e r s t a n d th e l im i t a t io n s o f th e s e p r o g r a m s , a l l o f t h e a s s u m p t io n s in h e r e n t w i th i n th e m m u s t

b e c lo s e ly e x a m in e d .

4 .1 M e a su r e m e n t E r r o r s

A c a re fu l f i e l d w o rk e r s h o u ld b e a b le t o c o l l e c t f a i r l y a c c u ra t e (± 5 ° o r l e s s ) o r i e n t a t i o n

d a t a fo r a p o p u la t i o n o f f a u l t s u s in g o n ly a c o m p a s s a n d c l i n o m e te r (C o m p to n , 1 9 6 2 , p . 2 1 -3 5 ;

R a g a n , 1 9 8 5 , p . 1 5 ) . T h e r e a r e , h o w e v e r , c o n d i t io n s u n d e r w h i c h s m a l l m e a s u r e m e n t e r r o r s

m a d e w h i l e d e t e r m i n i n g t h e s t r ik e o f a f a u l t p l a n e o r t h e t r e n d o f a l in e a t i o n w i t h i n t h a t fa u l t

p la n e m a y b e c o m e m a g n i f ie d .

W h e n , i n m e a s u r i n g t h e s t r ik e o f a f a u l t p l a n e , t h e c o m p a s s i s n o t h e l d e x a c t l y

h o r i z o n ta l t h e n a d i r e c t i o n o f s t r i k e w i l l b e m e a su re d o th e r th a n t h e t ru e s t r i k e . I f t h e a n g u la r

0d e p a r tu r e i n d e g r e e s o f th i s a p p a r e n t s t r i k e f ro m th e t ru e s t r i k e i s d e n o t e d a s , a n d th e d ip o f

St h e f a u l t p l a n e a s * , t h e n t h e r e s u l t a n t s t r i k e e r ro r (, ) m a y b e c a l c u l a t e d b y

T Equation 4.8 in Ragan (1985, p. 56) is given as: tan , = [tan(r+,) - tan *]cos * / 1 + [tan r tan(r+,)cos *]21

which is incorrect and has been corrected here in equation (2).

S 0, = s in [ t a n , / t a n * ] (1 )-1

F r o m t h i s e q u a t i o n ( R a g a n , 1 9 8 5 , p . 1 6 ) , i t m a y b e s e e n t h a t fo r s h a l l o w - d i p p i n g f a u l t

p l a n e s , t h e r e s u l t a n t s t r i k e e r r o r s m a y b e q u i t e l a r g e g i v e n f a i r l y s m a l l m e a s u r e m e n t e r r o r s

( f ig u re 4 -1 ) .

W h e n m e a s u r i n g t h e t re n d o f a l in e a t i o n o n a f a u l t p l a n e , i t i s a c o m m o n p r a c t i c e t o

a l i g n t h e c o m p a s s i n t h e d i r e c t i o n o f a p r o j e c t i o n o f th a t l i n e a t i o n o n t o a h o r i z o n t a l p l a n e . I f

a n e r ro r i s m a d e in th i s a l i g n m e n t , a s m e a su re d b y th e a n g le , , fo r a l i n e a t i o n o f p i t c h r o n a

Tf a u l t p l a n e o f d ip * , t h e n t h e r e s u l t a n t t r e n d e r ro r (, ) m a y b e c a l c u l a t e d b y

T, = t a n { [ t a n ( r+, ) - t a n ( r ) ] c o s * / 1 + [ ta n ( r ) t a n ( r+, ) c o s * ] } ( 2 ) -1 2

F r o m th i s e q u a t io n , i t m a y b e s e e n th a t fo r l i n e a t io n s w i th l a rg e p i t c h a n g l e s o n1

s t e e p l y - d i p p i n g f a u l t p l a n e s , a l a r g e t re n d e r ro r fo r th o s e l in e a t i o n s m a y o c c u r ( f i g u r e 4 - 2 ) .

I n a d d i t io n , t h e m a x i m u m e r ro r a s s o c i a t e d w i t h ( r -, ) i s l e s s t h a n i t i s fo r ( r +, ) a n d re p e a te d

m e a s u r e m e n t s w i l l n o t b e s y m m e t r ic a l l y d i s t r i b u t e d a r o u n d t h e t r u e t r e n d ( R a g a n , 1 9 8 5 , p . 5 6 ) .

G iv e n t h e a b o v e in f o rm a t io n , i t w o u ld b e u s e fu l to k n o w h o w s e n s i t i v e p a l e o s t r e s s

a n a l y s i s p r o g r a m s a r e t o s m a l l v a r i a t i o n s i n t h e o r i e n t a t i o n s o f t h e f a u l t p l a n e s a n d t h e i r s l i p

d i r e c t io n s . I f s m a l l m e a su re m e n t e r ro rs g a v e s ig n i f i c a n t ly d i f f e re n t r e s u l t s fo r th e c a lc u la t e d

p a l e o s t r e s s t e n s o r s , t h e s e p r o g r a m s w o u l d l o s e s o m e o f t h e i r u s e f u l n e s s g i v e n g e o l o g i c a l l y

re a l i s t i c fa u l t p o p u la t i o n d a ta .

F ig u r e 4 - 1 - G ra p h o f th e m a x im u m s t r i k e e r r o r f o r a f a u l t p l a n e a r i s in g f ro m a s t r i k e

0m e a s u r e m e n t e r r o r (, ) o f 1 ° to 5 ° a s a fu n c t io n o f th e d ip o f th e f a u l t p la n e ( f i g u re m o d i f i e d

f r o m R a g a n , 1 9 8 5 , p . 1 6 ) .

F ig u r e 4 - 2 - G ra p h o f th e m a x im u m t r e n d e r ro r o f a s l i c k e n l in e o n a f a u l t p l a n e a r i s in g f ro m

a t re n d m e a s u r e m e n t e r r o r (, ) o f 3 ° a s a f u n c t i o n o f t h e p i t c h o f t h e s l i c k e n l i n e ( 1 0 ° t o 8 0 ° i n

1 0 ° i n c r e m e n t s ) a n d t h e d i p o f t h e f a u l t p l a n e ( f i g u r e m o d i f ie d f ro m R a g a n , 1 9 8 5 , p . 5 7 ) .

4 .2 D e te r m in in g F a u l t S l ip

T h e d i r e c t io n o f s l i p o n a f a u l t i s c o m m o n ly o b ta in e d b y e x a m i n in g l i n e a t io n s k n o w n

a s s l i c k e n l in e s o n th e f a u l t ' s s u r f a c e (T j i a , 1 9 6 4 ; M e a n s , 1 9 8 7 ) . S l i c k e n l in e s a re l i n e a r

s t r i a t i o n s , o r g ro o v e s , r e s u l t i n g f ro m f r i c t i o n o r s h e a r s t r a in o n f a u l t s u r f a c e s a n d in d i c a t in g

t h e l a s t d i r e c t i o n o f m o v e m e n t o n t h a t f a u l t ( F l e u t y , 1 9 7 5 ) . I t i s p o s s i b l e f o r f a u l t s u r fa c e s t o

c o n t a i n m o r e t h a n o n e s e t o f s l i c k e n l in e s a n d c a re f u l e x a m in a t io n m a y b e n e e d e d to d i s t i n g u i s h

th e l a t e s t s l i p d i r e c t i o n f ro m e a r l i e r o n e s . A l s o , s l i c k e n l in e s a re a x ia l d a t a (C h e e n e y , 1 9 8 3 , p .

1 0 -1 1 ) , g iv in g tw o p o s s ib le s l i p d i r e c t i o n s 1 8 0 ° a p a r t , a n d so m e o th e r c r i t e r i a a r e th u s n e e d e d

t o e s t a b l i s h t h e s e n s e o f s l i p o f th e f a u l t ( i . e . w h ic h e n d o f a s l i c k e n l in e p o in t s i n th e d i r e c t i o n

o f m o v e m e n t o f th e o p p o s in g fa u l t b l o c k ) . T h e b e s t s e n s e -o f - s l i p in d i c a t o r s a r e g e o m e t r i c o r

p h y s i c a l l i n e s w h i c h h a v e b e e n o f f s e t b y f a u l t in g ( D a v i s , 1 9 8 4 , p . 2 6 8 ; R a g a n , 1 9 8 5 , p . 9 2 - 9 3 ) .

T h e s e f e a t u r e s m u s t b e u s e d c a u t i o u s l y a s s l i p d i r e c t i o n i n d i c a t o r s , h o w e v e r , s i n c e t h e y r e c o r d

th e n e t s l ip o n th e f a u l t w h ic h m a y b e th e r e s u l t o f s e v e r a l d i s t i n c t s l i p e v e n t s w i th d i f f e r in g

s l i p d i r e c t i o n s ( f i g u re 4 -3 ) .

B e f o r e 1 9 5 8 , i t w a s c o n s i d e r e d a x i o m a t i c t h a t s t e p - l ik e f e a t u r e s o n f a u l t p l a n e s c o u l d

b e u s e d a s s e n s e -o f - s l i p in d ic a to rs (H o b b s , e t . a l . , 1 9 7 6 , p . 3 0 4 ) . B y ru n n in g y o u r h a n d o v e r

th e f a u l t s u r f a c e , t h e d i r e c t i o n o f l e a s t r e s i s t a n c e ( i . e . t h e d i r e c t i o n w h e re y o u r h a n d ju m p s o v e r

th e r i s e r s o f th e s te p s r a t h e r t h a n s l a m m in g in to th e m ) i s t h e d i r e c t i o n o f m o v e m e n t o f th e

o p p o s in g fa u l t b lo c k ( f i g u re 4 -4 ) . A p ro b le m w i th t h i s m e th o d i s t h a t o th e r w o rk e rs h a v e s in c e

c l a im e d th a t s t e p s a re a n u n re l i a b l e s e n se -o f - s l i p in d i c a t o r s in c e s t e p s w i th a n in c o n g ru o u s

s e n se -o f - s l i p a re k n o w n f ro m th e f i e l d a n d th e l a b o ra to ry (P a t e r s o n , 1 9 5 8 ; T j i a , 1 9 6 4 ;

R ie c k e r , 1 9 6 5 ; T j i a , 1 9 6 7 ; N o r r i s a n d B a r r o n , 1 9 6 9 ; G a y , 1 9 7 0 ; H o b b s , e t . a l . , 1 9 7 6 , p . 3 0 3 -

3 0 5 ) . In 1 9 6 9 , N o r r i s a n d B a r ro n d i s t i n g u i s h e d b e tw e e n tw o d i f f e r e n t t y p e s o f s te p s fo u n d o n

fa u l t p l a n e s - - a c c r e t i o n s t e p s a n d f r a c t u re s te p s . A c c r e t io n s t e p s a r e fo rm e d b y th e a d h e s io n

o f m in e ra l i z e d g ou g e o n to th e s l i p su r f a c e a n d f r a c tu re s t e p s a re s t e p s w h ic h h a v e

F ig u r e 4 - 3 - D i a g r a m d e m o n s t r a t i n g h o w t h e n e t s l ip v e c t o r o n a f a u l t p l a n e m a y b e t h e r e s u l t

o f s e v e ra l d i s t i n c t s l i p e v e n ts w i t h d i f fe r i n g s l i p d i r e c t i o n s ( s l i p v e c to rs 1 th ro u g h 4 ) .

F ig u r e 4 - 4 - C r o s s - s e c t i o n a l v i e w o f a f a u l t p l a n e s h o w i n g h o w s t e p s m a y b e u s e d a s s e n s e - o f -

s l i p in d i c a t o r s o n f a u l t s u r f a c e s . T h e d i r e c t i o n o f l e a s t r e s i s ta n c e i s t a k e n t o b e th e d i r e c t i o n

o f m o t i o n fo r t h e o p p o s in g fa u l t b lo c k .

b e e n c u t in to th e s o l id ro c k . A c c r e t io n s t e p s a re fo rm e d a s th e s l i p s u r f a c e i s p a r t e d w i th th e

s te p 's r i s e r s f a c i n g p re f e r e n t i a l l y in th e d i r e c t i o n o f m o v e m e n t o f th e o p p o s in g b lo c k .

A c c r e t io n s t e p s a re t h e re fo re u s u a l ly c o n g ru o u s w i th th e f a u l t ' s s e n se -o f - s l i p . F r a c tu re s te p s

m a y f a c e i n e i t h e r d i r e c t i o n a n d t h u s m a y g i v e e i t h e r a c o n g r u o u s o r a n i n c o n g r u o u s s e n s e - o f -

s l i p . D u r n e y a n d R a m s a y (1 9 7 3 ) c l a im e d th a t a th i rd ty p e o f s t e p s fo rm e d f ro m la y e r s o f

f i b r o u s m in e ra l s o n th e s l i p s u r f a c e a l w a y s g a v e a c o n g ru o u s s e n se -o f - s l i p . T h e re fo re , w i t h

c a re , s t e p s m a y b e u se d a s s e n se -o f - s l i p in d i c a to rs o n s o m e fa u l t s (R o d , 1 9 6 6 ; T j i a , 1 9 6 7 ; T j i a ,

1 9 7 2 ; N o r r i s a n d B a r r o n , 1 9 6 9 ; D u rn e y a n d R a m s a y , 1 9 7 3 ; P e t i t , e t . a l . , 1 9 8 3 ; P e t i t , 1 9 8 7 ) .

O th e r p o s s i b l e s e n s e -o f - s l i p i n d i c a to r s a r e s t ru c tu r a l f e a tu r e s s u c h a s p r o d m a r k s ,

c r e s c e n t i c g o u g e s , p lu c k m a rk s , c h a t t e rm a rk s , p ro tu b e ra n c e s re s e m b l in g ro c h e s m o u to n n é e s ,

s p a l l s , b ru i s e d s t e p r i s e r s (T j i a , 1 9 6 7 ; T j i a , 1 9 7 2 ; G a m o n d , 1 9 8 3 ) , d r a g fo ld s (D a v i s , 1 9 8 4 ,

p . 2 7 0 -2 7 2 ; H o b b s , e t . a l . , 1 9 7 6 , p . 3 0 5 -3 0 6 ) , e n é ch e lo n t e n s io n g a s h e s , a n d th e o r i e n t a t i o n

o f a n y s e c o n d a ry s h e a r f r a c tu re s (G a m o n d , 1 9 8 3 ; P e t i t , e t . a l . , 1 9 8 3 ; H a n c o c k , 1 9 8 5 ; G a m o n d ,

1 9 8 7 ; P e t i t , 1 9 8 7 ) . F a u l t g o u g e (B y e r l e e , e t . a l . , 1 9 7 8 ) a n d s l ic k e n s id e s (L e e , 1 9 9 0 ) m a y a l s o

c o n ta in m ic ro s t r u c tu ra l s e n s e -o f - s l i p in d ic a to rs w h e n e x a m in e d p e t r o g ra p h ic a l l y .

W h e n c o l l e c t i n g fa u l t p o p u la t i o n d a ta fo r p a le o s t r e s s a n a l y s i s p ro g ra m s , a

r e c o m m e n d e d f in a l c h e c k o n t h e s e n s e - o f - s l i p d a t a i s t o s e e w h e t h e r th e y a r e a l l c o n s i s t e n t w i t h

o n e a n o th e r - - a s in g l e r e v e r s e f a u l t i n a p o p u la t i o n o f n o rm a l f a u l t s s h o u ld s ig n a l c a u t io n s in c e

i t i s u n l i k e ly to b e lo n g to th e s a m e s t r e s s f i e ld a s th e o th e rs .

4 .3 F a u l t M o r p h o lo g y

A n i m p l i c i t a s s u m p t i o n i n p a l e o s t r e s s a n a l y s i s i s th a t fa u l t s a r e p l a n a r ( i . e . t h e y m a y

b e d e s c r ib e d b y a u n iq u e s t r i k e , d ip , a n d d ip d i r e c t i o n ) . In r e a l i t y , h o w e v e r , f a u l t s a r e n o t

p e r f e c t l y p la n a r o n a n y s c a le (S c h o lz , 1 9 9 0 , p . 1 4 6 -1 4 7 ) .

M o s t f a u l t s s h o w a d e g re e o f c u rv a tu re in d ip s e c t i o n s o r i n p la n v ie w (M a n d l , 1 9 8 8 ,

p . 2 4 ) . L i s t r i c ( s h o v e l - s h a p e d ) n o r m a l a n d t h r u s t f a u l t s a r e e x t r e m e e x a m p l e s o f th i s a n d a r e

q u i t e c o m m o n in a re a s o f th in -s k in n e d t e c to n ic s . S u b s u r fa c e s t r e s s d i s t r i b u t io n s h a v e b e e n

u s e d to a c c o u n t fo r th e d e v e l o p m e n t o f l i s t r i c f a u l t s (H a fn e r , 1 9 5 1 ; J a ro s z e w s k i , 1 9 8 4 , p . 2 1 5 -

2 1 7 ) a n d f a c t o r s w h i c h m a y a f fe c t th e c u r v a t u r e o f a d e v e l o p i n g f a u l t i n c l u d e a n i s o t r o p i e s i n

t h e s h e a r i n g s t r e n g t h o f th e r o c k m a s s , c o m p a c t i o n b y t h e o v e r b u r d e n , a b n o r m a l l y h i g h p o r e

f lu i d p r e s s u r e s , a n d c h a n g e s i n t h e t e c to n ic s t r e s s f i e l d (M a n d l , 1 9 8 8 , p . 2 4 ) . F a u l t s m a y a l s o

b e c u rv e d , o r w a v y , o n a sm a l l e r s c a l e (G a m o n d , 1 9 8 3 ; J a ro s z e w s k i , 1 9 8 4 , p . 2 1 8 ; H a n c o c k ,

1 9 8 5 ) d u e p r i m a r i ly t o a n i s o t r o p i e s o f th e r o c k m a s s o r c h a n g e s i n t h e t e c t o n i c s t r e s s f i e l d

d u r i n g th e i r fo rm a t i o n .

C u rv e d f a u l t s w i l l y i e ld d i f f e r e n t s t r i k e a n d d ip o r i e n t a t i o n s , d e p e n d in g u p o n w h e re o n

t h e f a u l t 's s u r fa c e t h e m e a s u r e m e n t s a r e m a d e , r e s u l t in g i n p r o b l e m s f o r p a l e o s t r e s s a n a l y s i s

p ro g ra m s s im i la r t o t h o s e p re s e n te d b y m e a su re m e n t e r ro rs . A n o th e r p ro b le m w i th c u rv e d

fa u l t s (d e a l t w i th in m o re d e t a i l i n s e c t io n 4 .4 ) i s t h a t f a u l t c u rv a tu re , m o re s o t h a n th e

m a x i m u m r e s o l v e d s h e a r s t re s s , m a y d e t e r m i n e t h e f a u l t ' s s l i p d i r e c t i o n w h e n t h e f a u l t i s

re a c t i v a te d .

A n o th e r c h a ra c te r i s t i c o f f a u l t s i s t h e i r d i s c o n t in u i ty . F i e ld s tu d i e s h a v e s h o w n th a t

f a u l t s , a t a l l s c a le s , a re d i s c o n t in u o u s a n d c o n s i s t o f n u m e ro u s d i s c re t e s e g m e n ts (W a l l a c e ,

1 9 7 3 ; S e g a l l a n d P o l l a r d , 1 9 8 0 ; M a n d l , 1 9 8 7 ; M a n d l , 1 9 8 8 , p . 4 3 - 4 7 ; S c h o l z , 1 9 9 0 , p . 1 5 1 ) .

D i s c o n t i n u o u s f a u l t s h a v e b e e n m o d e l l e d a s a r ra y s o f r i g h t - o r l e f t - s t e p p i n g p a i r s o f e n é ch e lo n

s e g m e n ts (S e g a l l a n d P o l l a r d , 1 9 8 0 ) w h ic h h a v e p ro n o u n c e d d i f f e re n c e s in m e c h a n ic a l

b e h a v i o r f ro m c o n t in u o u s f a u l t s a n d w h ic h in f lu e n c e t h e i r s l i p d i r e c t i o n s w h e n s u b je c t e d to a

g iv e n s t r e s s f i e ld .

In 1 9 7 0 , T c h a l e n k o d e m o n s t r a t e d th a t t h e fo rm a t io n a n d e v o lu t io n o f s h e a r z o n e s

in v o lv e d id e n t i c a l c h a ra c t e r i s t i c s t a g e s in d e p e n d e n t o f th e i r s i z e . S h e a r z o n e s w e re th u s s h o w n

to b e s e l f - s im i l a r f ro m t h e s c a le o f s h e a r -b o x e x p e r im e n ts ( t e n s o f m i l l im e te r s ) t o e a r th q u a k e -

p r o d u c i n g f a u l t s (h u n d r e d s o f m e t e r s ) . S e l f - s i m i l a r i t y a t d i f fe r e n t s c a l e s i s a n i m p o r t a n t

c h a r a c t e r i s t i c o f a c l a s s o f f r a c t a l s ( M a n d e l b r o t , 1 9 7 7 ; M a n d e l b r o t , 1 9 8 3 ) , a n d s e v e r a l w o r k e r s

h a v e s in c e d e m o n s t r a t e d th a t f a u l t s u r f a c e s m a y b e d e s c r ib e d b y f r a c t a l g e o m e t ry (B ro w n a n d

S c h o lz , 1 9 8 5 ; S c h o lz a n d A v i l e s , 1 9 8 6 ; O k u b o a n d A k i , 1 9 8 7 ; P o w e r , e t . a l . , 1 9 8 7 ; S c h o lz ,

1 9 9 0 , p . 1 4 7 ) .

F a u l t s w h o s e s u r fa c e r o u g h n e s s i s b e s t d e s c r i b e d b y a f r a c t a l , o r H a u s d o r f f -

B e s i c o v i t c h , d i m e n s i o n m a y b e u s e d i n p a l e o s t r e s s a n a l y s i s i f t h e o r i e n t a t i o n o f t h e f a u l t 's s l i p

p la n e , r a th e r t h a n th e f a u l t p la n e i t s e l f , i s u se d (S c h o lz , 1 9 9 0 , p . 1 4 7 ) . T h e s l i p p l a n e i s d e f in e d

a s th e i d e a l i z e d p la n e u p o n w h ic h th e s l i p v e c to r l i e s a n d m a y b e v ie w e d a s th e r e g io n a l m e a n

o f th e a c t u a l f a u l t p l a n e ( f i g u re 4 -5 ) . D e t e rm in in g th i s s l i p p l a n e f ro m a s m a l l e x p o s u re o f th e

fa u l t p la n e in a n o u tc ro p m a y n o t a lw a y s b e p o s s ib le .

A n t i th e t i c f a u l t s a l s o p o s e a p r o b l e m f o r p a l e o s t r e s s a n a l y s i s . A n t i th e t i c f a u l t s

( J a ro sz e w s k i , 1 9 8 4 , p . 2 1 2 ; M a n d l , 1 9 8 8 , p . 4 7 ) a re m in o r f a u l t s w i th a s e n se -o f - s h e a r w h ic h

i s o p p o s i t e t o th e g e n e r a l d i r e c t i o n o f a n e x te rn a l ly im p o s e d sh e a r . W h e n m o v e m e n t o c c u r s o n

a n o rm a l l i s t r i c fa u l t , fo r e x a m p le , t h e i n c re a se d c u rv a tu re o f th e u p p e r p a r t o f th e f a u l t c a u se s

t h e f a u l t w a l l s to s e p a r a t e . S e c o n d - o r d e r a n t i th e t i c f a u l t s a r i s e t o a c c o m o d a t e t h i s c h a n g e i n

g e o m e t r y . S i n c e t h e s e f a u l t s a r i s e d u e t o a s e c o n d a r y s t r e s s f i e l d d e v e l o p i n g a r o u n d t h e m a i n

f a u l t , c a r e f u l f i e l d w o r k m u s t b e p e r fo r m e d t o b e s u r e t h a t a n t i th e t i c f a u l t s a r e n o t in c l u d e d i n

fa u l t p o p u l a t io n s u s e d fo r p a l e o s t re s s a n a l y s i s .

F i n a l ly , t h e r e a r e a s p e c ia l c l a s s o f f a u l t s w i th a ro t a t i o n a l c o m p o n e n t o f s l i p (D o n a th ,

1 9 6 2 ; D a v i s , 1 9 8 4 , p . 2 6 6 ; J a ro s z e w s k i , 1 9 8 4 , p . 1 4 6 ; R a g a n , 1 9 8 5 , p . 8 9 ; M a n d a l a n d

C h a k ra b o r ty , 1 9 8 9 ; T w is s a n d G e fe l l , 1 9 9 0 ) . S u c h f a u l t s d o n o t h a v e a c o n s ta n t s l i p d i r e c t i o n ,

o r s e n s e - o f - s l i p , a n d c a n n o t b e u s e d i n p a l e o s t r e s s a n a l y s i s p r o g r a m s ( f i g u r e 4 - 6 ) .

4 .4 T h e R e la t io n sh ip o f S h ea r S tr ess to F a u l t S l ip D ir ec t io n s

A fu n d a m e n ta l a s su m p t io n o f a l l p a l e o s t r e s s a n a ly s i s p ro g ra m s i s t h a t f a u l t s l i p a lw a y s

F ig u r e 4 - 5 - D ia g ra m d e m o n s t r a t i n g th e d i f f e re n c e b e tw e e n a f a u l t ' s s l i p p la n e a n d th e a c tu a l

f a u l t s u r fa c e w h i c h m a y n o t b e p l a n a r ( f i g u r e m o d i f ie d f ro m S c h o l z , 1 9 9 0 , p . 1 4 8 ) .

o c c u r s i n t h e m a x i m u m r e s o l v e d s h e a r s t re s s d i r e c t i o n o n t h e f a u l t p l a n e ( C a r e y a n d B r u n i e r ,

1 9 7 4 ; A rm i j o a n d C is te rn a s , 1 9 7 8 ; A n g e l i e r , 1 9 7 9 ; E tc h e c o p a r , e t . a l . , 1 9 8 1 ; A n g e l i e r , e t .

a l . , 1 9 8 2 ; V a s s e u r , e t . a l . , 1 9 8 3 ; A n g e l i e r , 1 9 8 4 ; G e p h a r t a n d F o r sy th , 1 9 8 4 ; M ic h a e l , 1 9 8 4 ;

R e c h e s , 1 9 8 7 ; A n g e l i e r , 1 9 8 9 ) . I t m a y b e s h o w n , h o w e v e r , t h a t u n d e r c e r t a i n c o n d i t io n s f a u l t s

m a y n o t a lw a y s s l i p in th e d i r e c t i o n o f t h e m a x im u m re s o lv e d s h e a r s t r e s s .

F o r fa u l t s u r f a c e s w i th l o n g -w a v e l e n g t h a s p e r i t i e s ( i . e . w a v y o r c u rv e d f a u l t s ) , s l i d in g

w i l l o c c u r a t s o m e sm a l l a n g le N t o a s l i p p l a n e w i th a n o rm a l fo rc e o f N , a s h e a r in g fo rc e o f

S , a n d a c o e f f i c i e n t o f f r i c t i o n o f : ( J a e g e r a n d C o o k , 1 9 7 9 , p . 5 5 ; S c h o lz , 1 9 9 0 , p . 5 2 ) . T w o

e q u a t io n s m a y b e d e r iv e d r e l a t i n g N a n d S to t h e n o rm a l fo rc e (n ) a n d th e s h e a r fo rc e ( s ) a c t i n g

u p o n t h e r a m p o f t h e a s p e r i ty a s f o l l o w s ( f i g u r e 4 - 7 )

n = N c o s (N ) + S s in (N ) (3 )

s = S c o s (N ) - N s in (N ) (4 )

A s s u m i n g a C o u l o m b f a i l u r e c r i te r i o n s u c h t h a t

s = : n (5 )

th e fo l l o w in g e q u a t io n r e s u l t s u p o n th e s u b s t i t u t i o n o f e q u a t io n s (3 ) a n d (4 ) in to e q u a t io n (5 )

S c o s (N ) - N s in (N ) = : [N c o s (N ) + S s in (N ) ] (6 )

F ig u r e 4 - 6 - T w o f a u l t t y p e s w i t h a r o t a t i o n a l c o m p o n e n t o f s l i p . A . H i n g e f a u l t s a n d B .

P iv o ta l f a u l t s ( f i g u re m o d i f ie d f ro m R a g a n , 1 9 8 5 , p . 8 9 ) .

F ig u r e 4 - 7 - C ro s s - s e c t io n a l v i e w o f a f a u l t s u r f a c e p a r a l l e l t o s h e a r in g w i th a n a s p e r i t y A

c re a t i n g a n a n g le N w i th th e f a u l t s u b j e c te d to a to ta l s h e a r in g fo rc e o f S a n d a to ta l n o rm a l

fo rc e o f N . T h e r a m p o f a s p e r i t y A i s s u b je c t e d to th e r e s o lv e d n o rm a l fo rc e (n ) a n d s h e a r in g

fo rc e ( s ) .

M u l t ip l y i n g e q u a t io n (6 ) b y [1 / c o s (N ) ] a n d a lg e b r a i c a l ly r e a r ra n g i n g y i e l d s

S = [: + t a n (N ) ] / [ 1 - : t a n (N ) ] N (7 )

w h ic h r e d u c e s to

S = : N (8 )

w h e n th e re a r e n o a s p e r i t i e s in th e s h e a r in g d i r e c t i o n (N = 0 ) . T h e r e f o r e , s h e a r i n g o n a f a u l t

s u r f a c e w i th a s p e r i t i e s w i l l r e q u i r e a l a rg e r s h e a r in g f o rc e (S ) t h a n s h e a r in g o n a fa u l t s u r f a c e

w i t h o u t a s p e r i t i e s f o r s u f f i c i e n t l y s m a l l v a l u e s o f N . T h e d i f f e r e n c e i s a re s u l t o f th e in c re a s e

in th e f r i c t i o n a l c o e f f i c i e n t t e rm o f e q u a t io n (8 ) f ro m : t o [: + t a n (N ) ] / [ 1 - : t a n (N ) ] o f

e q u a t i o n ( 7 ) p r o v i d e d t h a t N < t a n ( 1 / : ) s i n c e a t N = t a n ( 1 / : ) t h e e q u a t io n c h a n g e s s ig n .-1 -1

A s s u m e a n u n d u la t i n g fa u l t p la n e ( f i g u re 4 -8 ) s u c h th a t

S = : ' N (9 )

r e p re s e n ts s h e a r in g i n t h e m a x im u m r e s o lv e d s h e a r s t r e s s d i r e c t i o n a t s o m e a c u te a n g le " t o

t h e l o n g a x i s o f t h e u n d u l a t i o n s w h e r e

: ' = [: + t a n (N ) ] / [ 1 - : t a n (N ) ] (1 0 )

T h e e q u a t io n

F ig u r e 4 - 8 - A n u n d u la t i n g fa u l t p l a n e w i t h t h e m a x im u m re s o lv e d s h e a r s t r e s s d i r e c t i o n

(v e c to r S ) m a k in g a n a n g le " w i th t h e s l i p d i re c t i o n w h ic h c o r re s p o n d s t o t h e l o n g a x i s o f th e

u n d u la t i o n s (v e c t o r S ' ) . I t i s a s su m e d th a t o n ly s l i d in g p a ra l l e l t o th e l o n g a x i s o f th e

u n d u la t i o n s e n c o u n te rs n o a s p e r i t i e s .

S c o s (" ) = : N (1 1 )

S = : N / c o s (" ) (1 2 )

w i l l t h e n re p re s e n t s h e a r i n g p a ra l l e l t o th e lo n g a x i s o f t h e u n d u la t i o n s .

S e t t i n g e q u a t io n (9 ) e q u a l to e q u a t io n (1 2 ) y i e l d s

: ' N = : N / c o s (" ) (1 3 )

: ' = : / c o s (" ) (1 4 )

S u b s t i t u t in g e q u a t io n (1 0 ) in t o e q u a t io n (1 4 ) y i e l d s

[: + t a n (N ) ] / [ 1 - : t a n ) ] = : / c o s (" ) (1 5 )

w h ic h r e d u c e s , t h ro u g h a lg e b ra i c m a n ip u la t i o n , t o

N = t a n [: - : c o s (" ) ] / [ c o s (" ) + : ] (1 6 )-1 2

A s su m in g a g e o lo g ic a l ly r e a s o n a b le c o e f f i c i e n t o f f r i c t i o n (: ) o f 0 .8 5 fo r u p p e r c ru s t a l

ro c k s (B a r to n a n d C h o u b e y , 1 9 7 7 ; B y e r l e e , 1 9 7 8 ) , e q u a t io n (1 6 ) m a y b e re w r i t t e n a s

N = t a n [0 .8 5 - 0 .8 5 c o s (" ) ] / [ c o s (" ) + 0 .7 2 2 5 ] (1 7 )-1

a n d t h e r e l a t i o n s h i p b e tw e e n t h e a n g le s " a n d N m a y b e g r a p h e d a s " v a r i e s f ro m 0 ° t o 9 0 °

( f ig u re 4 -9 ) .

F ro m th e a b o v e , i t m a y b e s e e n t h a t n o n -p l a n a r f a u l t s ( i . e . m o s t f a u l t s ) w i th f a v o ra b ly -

a l ig n e d a s p e r i t i e s d o n o t a l w a y s s l i p i n th e d i re c t i o n o f th e i r m a x im u m re s o lv e d s h e a r s t r e s s .

4 .5 F a u l t in g P h a se D if f e re n t ia t io n

A n i m p o r t a n t p r o b l e m i n p a l e o s t r e s s a n a l y s i s u s i n g s t r ia t e d - fa u l t p o p u l a t i o n s i s

d e t e rm in in g i f e a c h o f t h e fa u l t s i n th e p o p u la t i o n w e re a c t i v a t e d w i th in a s in g l e s t r e s s f i e l d .

C a re fu l f i e l d w o rk c a n o f t e n d i s t i n g u i s h t h e r e l a t i v e a g e s o f f a u l t s u s in g c ro s s -c u t t i n g

re l a t i o n s h ip s b u t th e re i s n o g u a ra n t e e t h a t c o n t e m p o ra n e o u s ly - fo rm e d f a u l t s a l l b e lo n g to th e

s a m e fa u l t i n g p h a s e s in c e a f a u l t i s c r e a te d a t o n e t im e a n d m a y th e n b e r e a c t iv a te d m a n y t im e s

d u r i n g i t s e x i s t e n c e . P a l e o s t r e s s a n a ly s i s p ro g ra m s a s s u m e th a t f a u l t i n g w i l l o c c u r o n p re -

e x i s t i n g p la n e s o f w e a k n e s s (A n g e l i e r , 1 9 7 9 ; E tc h e c o p a r , e t . a l . , 1 9 8 1 ; A n g e l i e r , e t . a l . , 1 9 8 2 ;

V a s s e u r , e t . a l . , 1 9 8 3 ; A n g e l i e r , 1 9 8 4 ; G e p h a r t a n d F o r s y th , 1 9 8 4 ; M ic h a e l , 1 9 8 4 ; R e c h e s ,

1 9 8 7 ; A n g e l i e r , 1 9 8 9 ) . T h e re fo re , f a u l t s o f w id e ly d i s p a ra te in i t i a t io n a g e s m a y b e r e a c t iv a te d

w h e n s u b j e c t e d t o a g i v e n p a l e o s t r e s s f i e l d . O b s e r v i n g g e o m e t r ic r e l a t i o n s h i p s b e t w e e n f a u l t s

a n d th e p re s e n c e o f a n y s u c c e s s iv e s t r i a t i o n s o f d i f f e re n t a t t i t u d e s o n fa u l t s i n d i c a t e s th a t tw o

o r m o re p a l e o s t r e s s f i e l d s h a v e b e e n r e c o r d e d b u t , i n g e n e ra l , d i s t i n g u i s h in g w h ic h f a u l t s h a v e

b e e n r e a c t iv a t e d a t t h e s a m e t im e i s a d i f f i c u l t p ro b le m w h ic h h a s n o t b e e n w id e ly a d d re s se d .

I n t h e p a s t t e n y e a r s , n u m e r i c a l a l g o r i th m s h a v e b e e n p r o p o s e d w h i c h s e p a r a t e f a u l t s

i n t o d i f f e r e n t fa u l t in g p h a s e s ( A n g e l i e r a n d M a n o u s s i s , 1 9 8 0 ; H u a n g a n d A n g e l i e r , 1 9 8 7 ;

F ig u r e 4 - 9 - G ra p h o f th e c h a n g e in a lp h a (" ) f r o m 0 ° t o 9 0 ° f o r d i f fe r i n g a n g l e s o f p h i (N ) .

R e g i o n 1 c o n t a i n s t h e a n g l e s o f " a n d N f o r w h i c h s l i d i n g w i l l o c c u r p a r a l l e l to t h e l o n g a x i s

o f th e u n d u l a t i o n s a n d r e g i o n 2 c o n t a i n s t h e a n g l e s o f " a n d N f o r w h ic h s l i d i n g w i l l o c c u r

p a r a l l e l t o th e m a x i m u m r e s o l v e d s h e a r s t re s s d i r e c t i o n . F o r s m a l l a n g l e s o f " , e v e n v e r y s m a l l

l o n g - w a v e l e n g t h a s p e r i t i e s w i l l a c t a s b a r r i e r s to s l ip i n th e m a x im u m re s o lv e d s h e a r s t r e s s

d i r e c t i o n .

G a l in d o -Z a ld iv a r a n d G o n z a l e z - L o d e i ro , 1 9 8 8 ; H u a n g , 1 9 8 8 ) . U n fo r tu n a t e ly , t h e s e m e th o d s

a l l r e l y o n t h e s a m e c o m p u t a t i o n a l m e t h o d s u s e d t o d e t e r m i n e p a l e o s t r e s s a x e s f ro m f a u l t

p o p u la t i o n s (c h a p te rs 6 a n d 7 ) . S e p a ra t i n g fa u l t s in to h o m o g e n e o u s fa u l t in g p h a se su b se t s w i th

th e s e p r o g r a m s w i l l t h e re f o r e g u a ra n t e e a n e x c e l l e n t f i t fo r a p a l e o s t r e s s t e n s o r w h e n t h e s e

s u b s e t s a r e r u n t h r o u g h a n a n a l y s i s p r o g r a m . T h e r e s u l t s m a y n o t b e v e r y u s e f u l , h o w e v e r ,

s i n c e t h e s u b s e t s a r e e s s e n t i a l l y c r e a t e d b y h o w w e l l t h e y w i l l c o n s t r a i n a p a l e o s t r e s s t e n s o r .

4 .6 D e te rm in in g a P a le o str ess T e n so r

P a le o s t r e s s a n a ly s i s p ro g ra m s a t t e m p t to d e f in e a p a l e o s t r e s s f i e l d w h ic h i s c o n s i s te n t

w i th a p o p u l a t io n o f s t r i a t e d fa u l t s . T h e im p l ic a t i o n o f t h i s i s th a t a s i n g l e , u n i q u e p a l e o s t r e s s

t e n s o r i s r e c o r d e d i n t h e f a u l t in g r o c k s a t a s p e c i f ic p o i n t i n t i m e . A m a j o r p r o b l e m w i t h t h i s

i m p l i c a t i o n i s th a t s t re s s f i e l d s a r e n o t a l w a y s s t a t i c - - t h e y m a y e v o l v e w i t h t i m e ( M a n d l ,

1 9 8 8 , p . 1 5 -1 6 ) .

C o n s id e r tw o t h r u s t f a u l t s s i t u a t e d w i th in a s t r e s s f i e l d w h e re t h e l e a s t c o m p re s s iv e

3p rin c ip a l s t r e s s a x i s (F ) i s v e r t i c a l a n d c o n s t a n t i n m a g n i tu d e , t h e in t e rm e d ia t e p r inc ipa l s t re s s

2a x i s (F ) i s h o r i z o n t a l a n d c o n s ta n t in m a g n i tu d e , w h i l e t h e m o s t c o m p r e s s iv e p r in c ip a l s t r e s s

1a x i s (F ) i s h o r i z o n t a l a n d s t e a d i l y i n c r e a s i n g in m a g n i tu d e . I f t h e tw o fa u l t s h a v e a s t r i k e

1d i re c t io n p e rp e n d i c u l a r to F a n d d i f f e r o n ly in th e i r d ip a n g l e s , i t i s q u i t e p o s s ib le fo r th e m

b o th to b e a c t i v a t e d b y e s s e n t i a l l y th e s a m e s t r e s s f i e ld a t tw o d i f f e re n t t im e s a n d w i th tw o

d i f f e r e n t s l i p d i r e c t i o n s . T h e d i f f e r e n t s l i p d i r e c t i o n s a r i s e s in c e , a c c o rd in g to B o t t (1 9 5 9 ) , t h e

1s l i p d i r e c t i o n i s a fu n c t io n o f t h e r a t io s o f th e p r i n c i p a l s t r e s s e s a n d th i s ra t io c h a n g e s a s F

in c re a s e s in m a g n i t u d e ( f i g u re 4 -1 0 ) .

F a u l t s m a y a l s o t o t a l l y s w i t c h s ty l e d u r in g a s in g l e t e c t o n ic e v e n t (M a n d l , 1 9 8 8 , p . 1 5 -

1 6 ) . A s t h e p r in c ip a l s t r e s s a x e s in c re a s e o r d e c re a s e in m a g n i tu d e , t h e y m a y e x c h a n g e

3 2 1o r ie n t a t io n s ( i . e . a h o r i z o n t a l F i n c re a s e s i n m a g n i tu d e u n t i l i t b e c o m e s F a n d th e n F w h i l e

2 1 3 2F a n d F b e c o m e F a n d F r e s p e c t iv e ly ) . A s th e p r in c ip a l s t r e s s a x e s c h a n g e o r i e n t a t io n s ,

fa u l t in g s ty le s m a y s w i t c h b e tw e e n n o rm a l , w re n c h , o r t h ru s t f a u l t s .

A n o th e r w a y in w h ic h f a u l t s m a y c h a n g e i n s ty l e i s b y g ra v i t a t i o n a l r e - e q u i l i b ra t i o n .

A th ru s t f a u l t m a y , w h e n th e h o r i z o n ta l c o m p re s s iv e s t r e s s b e g in s l e s s e n in g , b e h a v e l i k e a

n o rm a l f a u l t (B e u tn e r , 1 9 7 2 ; J a ro s z e w s k i , 1 9 8 4 , p . 1 7 2 ) .

A c c o r d i n g to E d e lm a n ( 1 9 8 9 ) , w h i l e r e a s o n a b l e e s t im a t e s m a y b e m a d e f o r p a l e o s t r e s s

s t a t e s u s i n g s m a l l f a u l t s , l a r g e f a u l t s a r e i n d i c a t i v e o f l a r g e , f in i t e , n o n e l a s t i c s t r a i n s a n d t h e r e

a re n o c o n s t i t u t i v e e q u a t io n s r e l a t i n g s t r e s s a n d p e rm a n e n t s t r a i n . I f t h e s t r a in r a t e , c o a x i a l i t y ,

a n d v i s c o s i ty te n so r w e re k n o w n fo r s o m e in s t a n t in t im e in t h e d e fo rm a t io n h i s to ry o f a ro c k

m a s s , c a l c u la t i o n o f th e s t r e s s w o u ld b e t r i v ia l a n d c o n ta in t h e p ro p o g a te d e r ro rs o f th e o th e r

m e a s u r e m e n t s . In o t h e r w o r d s , a s i n g l e p a l e o s t r e s s d e t e r m i n a t i o n i s a d e r i v e d , u n v e r i f i a b l e

q u a n t i t y . T h i s i s im p o r t a n t s i n c e p a l e o s t r e s s a n a ly s i s p ro g ra m s a r e o f t e n u s e d to d e t e rm in e th e

p a l e o s t r e s s o r i e n t a t i o n s f o r v e r y l a r g e - s c a l e f a u l t s s u c h a s t h e S a n A n d r e a s a n d C o a l i n g a f a u l t

s y s te m s (M ic h a e l , 1 9 8 7 b ; J o n e s , 1 9 8 8 ) .

4 .7 D isc u ss io n

T o e v a lua te the pe r fo rm a n c e of p a leo s t res s a n a ly s i s a lg o r i th m s , I c re a te d a r t i f i c ia l f a u l t

p o p u la t i o n s u s in g th e s a m e in i t i a l a s s u m p t io n s th a t a re u s e d b y th e p ro g ra m s . T h e a r t i f i c i a l

f a u l t p o p u la t i o n s c o n s i s t o f p e r f e c t ly p l a n a r n o rm a l f a u l t s w i th a n e x a c t s t r i k e a n d d ip s i t u a te d

1 2 3w i th i n a s t a t i c s t r e s s f i e l d w i th t h e p r in c ip a l s t r e s s a x e s F , F , a n d F h a v in g o r i e n t a t i o n s o f

n o r th , u p , o r e a s t . T h e s l i p v e c to r s fo r e a c h f a u l t i n th e p o p u la t i o n a re c a l c u l a t e d u s in g th e

s a m e i n i t i a l a s s u m p t io n s t h a t p a l e o s t r e s s a n a l y s i s p ro g ra m s u s e w h e n c a l c u l a t in g p a l e o s t r e s s

t e n s o rs - - B o t t ' s fo rm u la (B o t t , 1 9 5 9 ) i s u t i l i z e d t o f i n d th e m a x im u m

F ig u r e 4 -1 0 - M o h r c i r c l e s d e m o n s t r a t i n g h o w tw o fa u l t s o f s l i g h t ly d i f f e r e n t o r i e n t a t i o n s

1(g r a p h e d a s 1 a n d 2 o n th e d i a g r a m s ) w i l l s l i p a t d i f f e r e n t t im e s a s F in c re a s e s . A . N o s l i p

1 1o c c u rs in i t i a l l y . B . F a u l t 1 b e g i n s to s l ip a s F in c re a s e s . C . B o t h f a u l t s 1 a n d 2 s l ip a s F

re a c h e s a m a x im u m v a lu e .

s h e a r s t r e s s d i r e c t i o n a n d th i s i s a s su m e d to b e i d e n t i c a l t o t h e f a u l t ' s s l i p d i r e c t i o n . R u n n in g

th e se fa u l t p o p u l a t i o n s t h ro u g h a p a le o s t r e s s a n a ly s i s p ro g ra m s h o u ld th u s g iv e t h e e x a c t

o r i e n t a t i o n s o f t h e p r i n c i p a l s t r e s s a x e s u s e d t o c r e a t e t h e m . I f t h e p a l e o s t r e s s a n a l y s i s

p ro g ra m s e x a m in e d re tu rn d i f fe re n t r e s u l t s , th e m a g n i t u d e o f t h e i r e r r o rs m a y b e e x a m in e d .

S y s t e m a t i c a l l y a l t e r in g th e o r i e n ta t i o n s o f th e i n d iv id u a l f a u l t s w i th in th e a r t i f i c i a l

f a u l t p o p u l a t io ns a l lo w s o n e to t e s t t h e s e n s i t i v i ty o f th e v a r io u s p a l e o s t r e s s a n a l y s i s p r o g r a m s .

A s a n e x a m p le , i n c re a s in g th e d ip o f o n e o f t h e f a u l t p la n e s in th e p o p u la t i o n b y a fe w d e g re e s

m a y b e v i e w e d a s e q u i v a l e n t to u s i n g a r e a l p o p u l a t i o n w h e r e t h e d i p o f o n e o f t h e f a u l t s i s

i n c o r re c t - - d u e e i t h e r to a m e a s u r e m e n t e r r o r o r p o s s i b l y t o t h e f a c t th a t th e f a u l t s u r fa c e i s

n o n -p la n a r . B y p la c in g a n e w fa u l t p la n e in to a p o p u la t i o n w i th a n a rb i t r a r i l y c h o s e n

o r i e n t a t i o n fo r i t s n o rm a l a n d s l i p v e c to r s , t h e s e n s i t i v i t y o f th e p a l e o s t r e s s a n a ly s i s p ro g ra m s

to th e a c c id e n ta l i n c lu s io n o f fa u l t s f ro m s e p a ra te t e c to n ic p h a s e s m a y b e e x a m in e d .

T h r o u g h c a r e f u l s e l e c t i o n o f t h e a r t i f i c i a l f a u l t p o p u la t io n s u s e d to t e s t t h e p a l e o s t r e s s

a n a l y s i s p r o g r a m s , t h e e f fe c t o f t h e v a r i o u s p r o b l e m s i n p a l e o s t r e s s a n a l y s i s d i s c u s s e d i n t h i s

c h a p te r m a y b e d e m o n s t r a t e d a n d e v e n q u a n t i f i e d to s o m e e x t e n t . T h a t i s th e a im o f th i s th e s i s .

C H A P T E R 5

G E N E R A T IN G A R T IF IC IA L F A U L T P O P U L A T IO N S

T o d e v e lo p s y n t h e t i c d a t a s e t s f o r t e s t in g c o m p u ta t io n a l m e t h o d s o f p a l e o s t r e s s

a n a ly s i s , I w ro te a P a s c a l p ro g ra m t o c a lc u la t e th e s l i p v e c to rs a n d th e s h e a r s t r e s s to n o rm a l

s t r e s s r a t i o s o n a n y a rb i t r a r i l y o r i e n t e d f a u l t p l a n e s i t u a t e d w i th in a s t r e s s f i e l d o f v a ry in g

p r i n c i p a l s t re s s m a g n i t u d e s . T h i s a l l o w e d m e t o d e r i v e a r t i f i c i a l d a t a s e t s o f f a u l t s a n d t h e i r

s l i p d i r e c t i o n s f o r a g i v e n s t r e s s t e n s o r . A c o m p l e t e l i s t i n g o f t h i s p r o g r a m i s g i v e n i n

A p p e n d ix C .

5 .1 T h e o r y

I t c a n b e s h o w n th a t fo r a g iv e n s t r e s s t e n s o r , t h e t o t a l s t r e s s a c t i n g u p o n a p la n e i n a n y

o r i e n t a t i o n w i th in th a t s t r e s s f i e l d m a y b e c a l c u l a t e d b y a re l a t i o n s h ip d e s c r ib e d b y M e a n s

(1 9 7 6 , p . 1 0 3 ) a s C a u c h y 's fo rm u la , w h ic h m a y b e w r i t t e n i n te n so r n o t a t i o n a s

i ij jT = F l (1 )

i jw h e re T re p re s e n t s t h e n o r th , u p , o r e a s t c o m p o n e n t s o f th e t o t a l s t r e s s v e c to r , l r e p re s e n t s t h e

ijn o r th , u p , o r e a s t d i re c t i o n c o s in e s (C h e e n e y , 1 9 8 3 , p . 1 1 2 ) , a n d F r e p re s e n t s e a c h o f th e n in e

c o m p o n e n t s o f t h e s t r e s s t e n so r . T h e n o r th , u p , a n d e a s t d i r e c t i o n c o s in e s a re d e f in e d h e re a s

th e c o s in e s o f th e a n g l e s b e tw e e n t h e f a u l t p l a n e n o r m a l v e c to r a n d th e n o r th , u p , a n d e a s t

c o o rd in a te a x e s w h ic h w i l l c o r r e s p o n d , fo r p u rp o s e s o f th i s s e c t io n , t o th e t h re e p r in c ip a l

1 2 3s t r e s s e s F , F , a n d F r e s p e c t i v e l y ( f ig u r e 5 - 1 ) .

W h e n t h e d i r e c t i o n c o s i n e s a r e c a l c u l a t e d f ro m t h e o r i e n t a t i o n o f t h e n o r m a l v e c t o r t o

th e f a u l t p la n e , C a u c h y 's fo rm u la m a y b e u se d to d e te rm in e th e c o m p o n e n t s o f th e t o ta l

F ig u r e 5 - 1 - R e l a t i o n s h i p b e t w e e n a p l a n e X Y Z s i t u a t e d w i t h i n a g e o g r a p h i c c o o r d i n a t e

1s y s t e m w h e re t h e n o r th , u p , a n d e a s t c o o rd i n a t e a x e s c o r re s p o n d t o th e p r in c ip a l s t r e s s a x e s F ,

2 3F , a n d F r e s p e c t iv e ly . T h e d i r e c t i o n c o s in e s a re th e c o s in e s o f th e a n g le s b e tw e e n th e p l a n e 's

1 2 3n o rm a l v e c t o r ( n ) a n d th e th r e e c o o r d i n a t e a x e s ( i . e . l = c o s (a lp h a ) , l = c o s (b e t a ) , a n d l =

c o s (g a m m a ) .

s t r e s s v e c to r a c t i n g u p o n th a t f a u l t p l a n e . S in c e th e p r in c ip a l s t r e s s a x e s h a v e th e s a m e

o r i e n t a t i o n a s th e n o r th , u p , a n d e a s t c o o rd in a te a x e s , t h e s h e a r s t r e s s t e rm s v a n i s h f ro m th e

s t r e s s t e n s o r a n d C a u c h y 's f o r m u l a ( e q u a t i o n 1 ) r e d u c e s t o

1 1 1T = F l

2 2 2T = F l (2 )

3 3 3T = F l

w h e re t h e 1 , 2 , a n d 3 c o m p o n e n t s a r e t h e n o r th , u p , a n d e a s t c o m p o n e n t s r e s p e c t iv e ly .

T o c a lc u la t e th e n o rm a l s t r e s s a n d sh e a r s t r e s s m a g n i tu d e s a c t i n g u p o n th e f a u l t p la n e ,

t h e a n g l e b e tw e e n t h e t o t a l s t r e s s v e c to r a n d th e n o rm a l v e c to r t o t h e f a u l t p l a n e m u s t b e fo u n d .

A s im p le re l a t i o n s h ip e x i s t s b e tw e e n a n y tw o v e c to rs in sp a c e a n d th e a n g le (2) b e tw e e n th e m

s u c h th a t

2 = c o s [ (a @ b ) / (2a 2 2b 2) ] (3 )-1

w h e re a @ b i s t h e i n n e r , o r d o t , p ro d u c t a n d 2a 2 a n d 2b 2 a r e th e m a g n i tu d e s o f v e c to r s a a n d b

r e s p e c t i v e l y ( M a r s d e n a n d T r o m b a , 1 9 8 1 , p . 2 0 ) . T h e i n n e r p r o d u c t b e t w e e n a n y t w o v e c t o r s

i s d e f in e d a s

1 1 2 2 3 3a @ b = a b + a b + a b (4 )

a n d t h e m a g n i tu d e o f a n y v e c t o r i s d e f in e d a s

1 2 32a 2 = ( a + a + a ) (5 )2 2 2 1/2

O n c e t h e a n g l e b e t w e e n t h e t w o v e c t o r s h a s b e e n c a l c u l a t e d , i t c a n b e s h o w n b y s i m p l e

n st r i g o n o m e t ry ( f i g u re 5 -2 ) th a t t h e n o r m a l s t r e s s (F ) a n d s h e a r s t r e s s (F ) m a g n i t u d e s a r e

nF = 2T 2 c o s (2) (6 )

sF = 2T 2 s i n (2) (7 )

O n c e t h e n o r m a l s t re s s a n d s h e a r s t re s s m a g n i t u d e s a c t i n g u p o n t h e f a u l t p l a n e a r e

s nc a l c u l a t e d , t h e f a u l t p l a n e ' s s h e a r s t r e s s to n o rm a l s t r e s s r a t i o (F / F ) m a y b e d e te rm in e d .

5 .2 D e r iv in g B o t t ' s F o r m u la

F o r a fa u l t p la n e o f a n y o r i e n ta t i o n w i th in a s t r e s s f i e ld , t h e re w i l l b e a m a x im u m s h e a r

s t r e s s d i re c t i o n a lo n g w h ic h s l i p m a y o c c u r . B o t t (1 9 5 9 ) s h o w e d th a t d i f f e r e n t r e l a t i v e

m a g n i tu d e s o f th e p r in c ip a l s t r e s s e s w i l l r e s u l t i n d i f f e r e n t d i re c t i o n s o f m a x im u m s h e a r s t r e s s .

I f t h e o r i e n t a t i o n o f th e f a u l t p l a n e r e l a t i v e t o t h e p r in c ip a l s t r e s s a x e s i s k n o w n , th e r e l a t i v e

m a g n i tu d e s o f th e p r in c ip a l s t r e s s e s c a n b e u s e d to d e t e r m i n e t h e m a x im u m s h e a r s t r e s s

d i r e c t i o n w i t h in th e fa u l t p la n e .

T o d e r iv e t h i s e q u a t io n , a s su m e a p l a n e X Y Z o f u n i t a re a s i t u a t e d w i th in a n o r th , u p ,

1 2 3a n d e a s t c o o r d i n a t e s y s t e m ( f ig u re 5 -3 ) . T h e p r in c i p a l s t r e s s a x e s F , F , a n d F a n d t h e i r

1 2 3a s so c i a t e d d i r e c t i o n c o s in e s l , l , a n d l c o in c id e w i th th e n o r th , u p , a n d e a s t d i r e c t i o n s

re s p e c t iv e ly . S in c e th e X Y Z p la n e i s o f u n i t a re a ,

1 2 3l + l + l = 1 (8 )2 2 2

1 2 3a n d l i s t h e a re a o f th e O Y Z p la n e , l i s th e a re a o f th e O X Z p la n e , a n d l i s t h e a re a o f th e

F ig u r e 5 - 2 - D e t e r m i n i n g t h e s h e a r s t re s s a n d n o r m a l s t re s s m a g n i t u d e s f ro m t h e a n g l e

b e t w e e n t h e t o t a l s t re s s v e c t o r a c t i n g u p o n a f a u l t p l a n e a n d t h e f a u l t p l a n e 's n o r m a l v e c t o r .

F ig u r e 5 - 3 - R e l a t i o n s h i p b e t w e e n a p l a n e X Y Z s i t u a t e d w i t h i n a g e o g r a p h i c c o o r d i n a t e

1 2 3 1s y s t e m w h e re t h e p r in c ip a l s t r e s s a x e s F , F , a n d F a n d t h e i r a s s o c i a t e d d i re c t i o n c o s in e s l ,

2 3l , a n d l c o r r e s p o n d to th e n o r t h , u p , a n d e a s t c o o rd in a te a x e s re s p e c t i v e ly .

1 1O X Y p l a n e . T h e re f o r e , t h e n o r m a l f o rc e o n th e O Y Z p la n e i s l F , t h e n o rm a l fo rc e o n th e O X Z

2 2 3 3p l a n e i s l F , a n d t h e n o r m a l fo r c e o n t h e O X Y p l a n e i s l F .

S in c e th e s y s t e m i s in e q u i l i b r iu m , th e th re e c o m p o n e n t s o f fo rc e a c t i n g u p o n th e X Y Z

1 1 2 2 3 3 Tp l a n e a r e ( - l F , - l F , - l F ) . T h e to ta l r e s u l t a n t fo rc e (F ) o n th e X Y Z p la n e i s t h e re fo re

T 1 1 2 2 3 3F = - ( l F + l F + l F ) (9 )2 2 2 2 2 2 1/2

Na n d th e n o rm a l f o rc e (F ) a c t i n g u p o n th e X Y Z p la n e , w h ic h m a y b e d e t e rm in e d b y re s o lv in g

e a c h c o m p o n e n t a l o n g t h e n o r m a l d i re c t i o n , i s

N 1 1 2 2 3 3F = - ( l F + l F + l F ) (1 0 )2 2 2

ST h e m a x im u m s h e a r fo rc e (F ) a c t i n g u p o n th e X Y Z p la n e m a y th e n b e c a l c u l a t e d u s in g

th e r e l a t i o n s h ip

T N SF = F + F (1 1 )2 2 2

w h i c h , u p o n s u b s t i t u t i o n o f e q u a t i o n s ( 9 ) a n d ( 1 0 ) in t o e q u a t i o n ( 1 1 ) , i s

S 1 1 2 2 3 3 1 1 2 2 3 3F = [ l F + l F + l F - ( l F + l F + l F ) ] (1 2 )2 2 2 2 2 2 2 2 2 2 1/2

w h i c h y i e l d s

S 1 1 3 2 2 3 1 1 3 2 2 3F = l (F - F ) + l (F - F ) - [ l (F - F ) + l (F - F ) ] (1 3 )2 2 2 2 2 2 2

a f t e r u s in g th e id e n t i ty in e q u a t i o n (8 ) a n d a lg e b ra ic a l l y re a r r a n g in g .

ST h e n e x t s t e p i s t o c a l c u l a t e t h e r e s o l v e d c o m p o n e n t s o f t h e m a x i m u m s h e a r fo r c e ( F )

S - strike S - dipa lo n g th e s t r i k e d i r e c t i o n (F ) a n d th e d ip d i r e c t i o n (F ) u s in g th e r e l a t i o n s h ip

S S - strike S - dipF = F + F (1 4 )2 2 2

R e s o lv i n g t h e c o m p o n e n t s o f s h e a r fo r c e a l o n g th e n o r th - e a s t d i re c t i o n y i e l d s

S - strike 1 3 1 1 3 1 3 3 1 3F = [ l l F / ( l + l ) ] - [ l l F / ( l + l ) ] (1 5 )2 2 1/2 2 2 1/2

S - strike 1 3 1 3 1 3F = [ l l (F - F ) / ( l + l ) ] (1 6 )2 2 1/2

S u b s t i t u t i n g e q u a t io n s (1 3 ) a n d (1 6 ) in to e q u a t io n (1 4 ) r e s u l t s i n

S - dip 1 1 3 2 2 3 1 1 3 2 2 3F = { l (F - F ) + l (F - F ) - [ l (F - F ) + l (F - F ) ] } - 2 2 2 2 2 2 2 2

1 3 1 3 1 3 [ l l (F - F ) / ( l + l ) ] (1 7 )2 2 1/2 2

w h i c h y i e l d s

S - dip 2 1 1 3 2 2 3 2F = l [ l (F - F ) - ( 1 - l ) (F - F ) ] / (1 - l ) (1 8 )2 2 2 1/2

a f t e r u s in g th e id e n t i ty in e q u a t i o n (8 ) a n d a lg e b ra ic a l l y re a r r a n g in g .

T h e p i t c h (" ) o f th e m a x im u m s h e a r s t r e s s v e c to r w i th in t h e X Y Z p la n e ( i . e . t h e a n g l e

b e tw e e n t h e s t r i k e o f th e X Y Z p la n e a n d t h e m a x im u m s h e a r s t r e s s d i r e c t i o n ) i s g iv e n b y

S - dip S - striket a n (" ) = F / F (1 9 )

w h i c h , u p o n s u b s t i t u t io n o f e q u a t io n s (1 6 ) a n d ( 1 8 ) , y i e l d s

2 1 1 3 2 2 3 2t a n (" ) = { l [ l (F - F ) - ( 1 - l ) (F - F ) ] / ( 1 - l ) } / 2 2 2 1/2

1 3 1 3 1 3 [ l l (F - F ) / ( l + l ) ] (2 0 )2 2 1/2

S u i ta b l e a lg e b ra i c r e a r r a n g e m e n t o f e q u a t io n (2 0 ) w i l l r e s u l t i n

1 2 2 2 1 3" = t a n [ l l - M l + M l ) / ( l l ) ] (2 1 )-1 2 3

w h e re M i s a u s e fu l v a l u e d e f in e d b y A n g e l i e r (1 9 7 9 ) a s

2 3 1 3M = (F - F ) / (F - F ) (2 2 )

w h ic h r a n g e s f ro m 0 .0 to 1 .0 a n d re p re s e n ts th e r e l a t i v e m a g n i tu d e s o f t h e p r in c ip a l s t r e s s e s

( i . e . r e p re s e n ts th e s h a p e o f t h e s t r e s s e l l ip s o id ) .

B o t t ' s fo rm u la (e q u a t io n 2 1 ) y ie ld s th e p i t c h a n g le o f t h e s l i p v e c to r o n th e f a u l t p la n e .

F o r p l o t t in g o n a s t e r e o g r a p h i c p r o j e c t i o n , i t m a y b e m o r e c o n v e n i e n t to r e p r e s e n t t h e s l i p

v e c to r s im p ly b y i t s p lu n g e a n d t r e n d . T h i s m a y b e d o n e u t i l i z in g th e t h re e fo rm u la s

b e t a = t a n [ t a n (p i t c h ) c o s (d ip ) ] (2 3 )-1

t r e n d = s t r i k e + b e t a (2 4 )

p lu n g e = c o s [c o s (p i t c h ) / c o s (b e t a ) ] (2 5 )-1

w h e re t h e p i t c h a n g le o f t h e s l i p v e c to r in th e f a u l t p la n e a n d th e d ip o f th e f a u l t p la n e a re u s e d

to c a l c u l a t e a v a lu e fo r b e t a w h ic h i s t h e h o r i z o n t a l a n g l e b e tw e e n th e t r e n d o f th e s l i p v e c to r

a n d th e s t r i k e o f th e f a u l t p la n e . T h e p lu n g e o f t h e s l i p v e c to r i s t h e n c a lc u la t e d f ro m th e p i t c h

a n g le a n d b e ta (R a g a n , 1 9 8 5 , p . 5 1 ) .

1 2 3W ith so m e m o d i f i c a t i o n s o f th e a b o v e e q u a t io n s , t h e F , F , a n d F a x e s m a y h a v e a n y

a rb i t r a ry o r i e n t a t i o n s o th e r th a n th e n o r th , u p , a n d e a s t o n e s a s s ig n e d to th e m . T o d o so , t h e

1 2 3l , l , a n d l d i r e c t io n c o s in e s m u s t r e p re s e n t th e c o s in e s o f th e a n g le s b e tw e e n th e n o rm a l

1 2 3v e c to r t o t h e f a u l t p l a n e a n d th e F , F , a n d F a x e s r e s p e c t iv e ly . T h ro u g h C a u c h y 's fo rm u la ,

1 2 3 1 2 3t h e T , T , a n d T to ta l s t r e s s v e c to r c o m p o n e n t s w i l l b e r e s o lv e d p a ra l l e l t o th e F , F , a n d F

a x e s . B o t t ' s fo rm u la (e q u a t io n 2 1 ) m u s t b e m o d i f i e d a c c o rd in g ly a n d e a c h o f th e s p e c i a l c a s e s

w h e r e t h e f a u l t p l a n e m a y b e p a r a l l e l to o n e o f t h e p r i n c i p a l p l a n e s m u s t b e d e a l t w i t h

in d iv id u a l ly . T h e r e s u l t a n t p i t c h a n g l e w i l l t h e n b e th e a n g l e b e tw e e n t h e i n t e r s e c t io n o f th e

1 3p la n e c o n t a in in g F a n d F w i th th e f a u l t p l a n e a n d th e s l i p v e c to r w i th in th a t f a u l t p l a n e

( f i g u r e 5 - 4 ) .

5 .3 P r o g r a m In p u t

A p ro g ra m w a s w r i t t e n in T u rb o P a s c a l v e r s io n 3 .0 1 to p e r fo rm th e a b o v e c a l c u l a t i o n s

a n d g r a p h i c a l l y d i s p l a y t h e r e s u l t s . T h e p r o g r a m r e q u i r e s n i n e i t e m s o f in f o r m a t i o n t o p e r fo r m

1 3t h e c a l c u l a t i o n s - - t h e o r i e n t a t i o n s o f F a n d F r e l a t i v e t o t h e n o r t h , e a s t , a n d u p c o o r d i n a t e

1 3a x e s , t h e m a g n i t u d e s o f F a n d F , t h e c o e f f i c i e n t o f f r i c t i o n (: ) o f t h e f a u l t

0p la n e , t h e c o h e s i o n (C ) o f th e f a u l t p la n e , t h e p lu n g e a n d t r e n d o f th e f a u l t p la n e 's n o rm a l

v e c t o r i n d e g r e e s , a n d t h e n u m b e r o f M v a lu e s to e x a m in e .

T h i s in fo rm a t io n i s o b t a in e d in t e r a c t iv e ly b y th e p ro g ra m a s t h e u s e r e n t e r s e a c h v a l u e

w h e n p ro m p te d ( f i g u re 5 -5 ) .

1 3T h e e n t e r e d m a g n i t u d e s o f F a n d F h a v e a rb i t r a ry u n i t s s in c e o n ly th e i r r e l a t i v e

1 2 3v a lu e s a re im p o r t a n t , n o t th e i r a b s o lu te v a lu e s . T h e p ro g ra m a l s o re q u i r e s th e F , F , a n d F

a x e s t o c o r re s p o n d t o e i t h e r th e n o r t h , u p , o r e a s t c o o r d i n a t e a x e s . T h i s i s d o n e o n l y t o s i m p l i fy

t h e m a t h e m a t i c s i n v o l v e d i n c a l c u l a t i n g t h e s l i p v e c t o r o r i e n t a t i o n s a n d d o e s n o t s i g n i f ic a n t l y

l i m i t th e p ro g ra m .

T h e u s e r i s r e q u i re d to e n t e r t h e a l p h a n u m e r ic c h a ra c t e r s " N " , " E " , o r " U " fo r th e

p r in c ip a l s t r e s s o r i e n t a t io n s , a r e a l n u m b e r b e tw e e n - 1 0 0 .0 a n d + 1 0 0 .0 f o r t h e p r in c ip a l s t r e s s

m a g n i tu d e s , a r e a l n u m b e r b e tw e e n 0 .0 a n d 1 0 0 .0 fo r th e c o e f f i e n t o f f r i c t i o n a n d th e c o h e s io n ,

a re a l n u m b e r b e tw e e n 0 .0 a n d 9 0 .0 fo r th e p lu n g e o f th e n o rm a l to th e f a u l t p la n e , a re a l

n u m b e r b e tw e e n 0 .0 a n d 3 6 0 .0 fo r t h e t r e n d o f t h e n o rm a l t o t h e f a u l t p l a n e , a n d a n i n t e g e r

2b e t w e e n 2 a n d 5 0 f o r t h e n u m b e r o f F in te rv a l s to e x a m in e .

O n c e t h e i n i t i a l d a t a h a s b e e n e n t e r e d , t h e p ro g ra m m a y b e g in to c a l c u l a t e s l i p v e c to r

o r i e n ta t i o n s fo r th e s p e c i f i e d f a u l t p la n e .

5 .4 P r o g r a m P r o c e d u r e s

T h e s l i p v e c t o r c a l c u l a t i o n p r o g r a m c o n s i s t s o f m a n y p r o c e d u r e s t o i n t e r a c t i v e l y i n p u t ,

c a l c u l a t e , a n d o u t p u t d a t a . M o s t o f t h e p r o g r a m p r o c e d u r e s i n A p p e n d i x C a r e t h e r e s i m p l y t o

e n a b l e t h e p ro g ra m to fu n c t io n in t e r a c t iv e ly , t o g ra p h i c a l ly d i s p la y th e r e s u l t s o f th e

c a l c u l a t i o n s , a n d t o c r e a t e t h e A u t o C A D D X F f i l e s . T h e o n l y p r o c e d u r e s I w i l l d i s c u s s h e r e

a r e t h o s e d i re c t l y i n v o lv e d w i th m a t h e m a t i c a l l y c a l c u l a t in g th e s l i p v e c t o r o r i e n t a t io n s .

F ig u r e 5 - 4 - L o w e r -h e m is p h e re s t e re o g ra p h ic p ro je c t io n s h o w in g th re e a rb i t r a r i l y o r i e n te d

1 2 3p r in c ip a l s t r e s s e s F , F , a n d F i n r e l a t i o n t o a n a rb i t r a r i l y o r i e n te d f a u l t p la n e w i th a n o rm a l

1 2 3v e c t o r n . A , B , a n d ' a r e t h e a n g l e s b e tw e e n t h e n o rm a l v e c to r a n d F , F , a n d F r e s p e c t iv e ly .

T h e p i t c h a n g l e s o f a n y s l i p v e c to r s w i l l b e m e a s u re d f ro m th e in t e r s e c t io n P o f t h e p la n e

1 3c o n t a in in g F a n d F w i t h th e fa u l t p la n e ( f i g u re m o d i f ie d f ro m S c h im m ri c h , 1 9 9 0 ) .

F ig u r e 5 - 5 - In te ra c t iv e sc re e n d i sp la ye d b y th e s l ip v e c to r c a l c u l a t io n p r o g ra m a s th e u se r

1e n te rs the in i t i a l d a ta . In th i s e x a m p le , th e o r ie n ta t io n o f F i s no r th w i th a m a g n i tu d e o f + 1 .0 ,

3th e o r ie n t a t i o n o f F i s e a s t w i th a m a g n i tu d e o f -1 .0 , th e c o e f f ic ie n t o f f r ic t io n is 0 .8 5 , th e

c o he s io n i s 0 .0 , th e fa u l t p la ne h a s a n o rm a l v e c to r o r ie n te d a t 7 0 /0 3 0 , a n d 2 1 M v a lu e s w i l l b e

2e xa m in e d ( s in c e 2 0 F in te rv a ls e q u a ls 2 1 M va lue s ) .

+))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),

* SLIP VECTOR PLOTTING PROGRAM ** ** ** ** What is the orientation of the maximum compressive ** principal stress axis F1 (North, East, or Up) ? N ** ** ** What is the orientation of the minimum compressive ** principal stress axis F3 (North, East, or Up) ? E ** ** ** Enter the value for F1 : 1.0 ** ** Enter the value for F3 : -1.0 ** ** ** Enter the coefficient of friction (:) : 0.85 ** ** Enter the cohesion (C) : 0.0 ** ** ** Now enter the plunge and trend of the normal ** vector to the fault plane you wish to examine ** ** Enter the plunge : 70 ** ** Enter the trend : 030 ** ** ** How many values of F2 between F1 ** and F3 do you wish to examine ? 20 ** *.)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))-

O n c e th e d a ta ha s b e e n in te ra c t iv e l y e n te re d th ro u g h th e p r o g ra m p r o c e d u re A sk D a ta ,

1 3th e o r ie n ta t io n o f th e F a n d F p r in c ip a l s t r e s s a x e s a n d th e p lu n g e a n d t r e n d o f th e fa u l t p la n e 's

n o rm a l v e c to r a re p a s se d to p r o c e d u re D irC o s in e s w h ic h c a lc u la te s th e th re e d i r e c t io n c o s in e s

1 2 3l , l , an d l u s ing the eq ua t io ns

nl = [ c o s ( p lu n g e ) c o s ( t r e n d ) ]

ul = s in (p lu n g e ) (2 6 )

el = [ c o s ( p lu n g e ) s in ( t r e n d ) ]

n u ew h e r e l , l , a n d l a r e th e d i r e c t io n c o s in e s re la t in g t h e fa u l t p l a n e 's n o r m a l v e c to r to th e n o r th ,

u p , an d e a s t c o o rd ina te a x e s re sp e c t ive ly ( C h e e n e y , 1 9 8 3 , p . 1 1 2 ) . T h e se m u s t b e c o n v e r te d to

1 2 3 1 2th e p r o p e r l , l , a n d l d i r e c t io n c o s in e s d e p e n d in g u p o n th e g iv e n o r ie n ta t io n s o f th e F , F , a n d

3F p r in c ip a l s t re ss a x e s .

1 3T h e d i r e c t io n c o s in e s a re u se d , a lo n g w i th t h e F a n d F m a g n i tu d e s , to c a lc u la te th e

th re e c o m p o n e n ts o f th e to ta l s t r e ss v ec to r ac t in g u p o n th e fa u l t p la n e b y p r o c e d u re C a u c h y .

2 3T h is p ro c e d u re in i t i a l ly se t s the m a g n i tud e o f F e q u a l to the m a g n i tud e o f F a nd th e n in c re a se s

1i t th ro u g h th e u se r -sp e c i f ie d n u m b e r o f s t e p s u n t i l i t i s e q u a l in m a g n i tu d e to F . F o r e a ch o f

1 2 3the se s te p s , the m a g n i tud e s o f F , F , a n d F a r e u se d to c a lc u la te a v a lu e fo r M ( eq u a t io n 2 2 )

a n d C a u c h y 's fo r m u l a i s u s e d to c a lc u la te th e to ta l s t r e ss v e c to r ( e q u a t io n 2 ) fo r e a c h M v a lu e .

T h is d a ta i s s to re d in tw o -d im e n s io n a l a r ra ys .

A l l o f th e d a ta is th e n p a sse d to p r o c e d u re C a lc u la te S tre sse s w h ich c a lcu la te s the s l ip

v e c to r p i tc h a n g le a n d th e s h e a r s t r e ss to n o r m a l s t r e s s r a t io fo r e a c h M v a lu e . T h e p r in c ip a l

2 3s t r e s s F i s o n c e a g a in se t eq u a l to the m a g n i tud e o f F a n d inc re a se d b y s te p s u n t i l i t i s eq u a l in

1m a gn i tu d e to F . C a lc u la t io n s a r e th e n p e r fo r m e d fo r e a c h M va lue .

T h e th r e e d i re c t io n c o s in e s a n d c o m p o n e n ts o f th e to ta l s t re ss v e c to r a re f i r s t use d to

c a lc u la te th e a ng le b e tw ee n th e fa u l t p l a ne 's n o rm a l ve c to r a n d th e to ta l s t r e ss ve c to r a c t in g up o n

the fau l t u s ing eq ua t io ns (3 ) , (4 ) , a n d ( 5 ) . F ro m th i s ang le , the shea r s t r e s s an d no rm a l s t r e s s

a c t in g u p o n th e fa u l t p la n e m a y b e c a lc u la te d u s in g e q u a t io n s (6 ) a n d (7 ) . T h e e ffe c t iv e sh e a r

s ns t r e ss to no rm a l s t r e ss r a t io (F ' / F ' ) i s th e n d e te rm in e d u s in g th e fo l lo w in g e q u a t io n

s n s n 0(F ' / F ' ) = F / [ (: F ) + C ] (2 7 )

np ro v id e d tha t : o r F a r e n o t e q u a l to z e r o . T h is e q u a t io n is u s e d s in c e a C o u lo m b fa i lu r e

c r i t e r io n i s as su m e d (C o u lo m b , 1 7 7 6 ; H a n d in , 1 9 6 9 ) .

T h e p ro g ra m n e x t c h e ck s fo r th e sp e c ia l c a se s w h e re th e fa u l t p la ne i s p a ra l le l to o ne o f

th e p r in c ip a l s t r e ss a x e s s in c e th is w i l l r e su l t in a d iv is io n b y z e r o i n p r o c e d u r e

C a lc u la te S tre sse s ( a fa u l t p l a n e p a r a l le l to a p r in c ip a l s t r e ss a x is w i l l h a v e a d i r e c t io n c o s in e

e q u a l to z e ro ) .

F ina l ly , B o t t 's fo rm u la (e q u a t io n 2 1 ) i s u se d to c a lcu la te the p i tch a n g le o f th e fau l t 's s l ip

v e c to r fo r e a c h M va lue . A s im p l i f i ed f lo w -ch a r t o f th e se p ro c e d u re s i s sh o w n in f igu re 5 -6 .

T h e r e su l t s o f th e s e c a lc u la t io n s a r e th e n n u m e r ic a l ly o r g r a p h i c a l ly d i sp la y e d s h o w i n g

the s l ip v e c to r o r ie n ta t io n s o n the us e r - sp e c i f ie d fau l t p lan e fo r e a c h va lue o f M c a lc u la te d a n d

the a sso c ia ted sh e a r s t re ss to no rm a l s t re ss r a t io s .

5 .5 P r o g r a m O u tp u t

T h e r e a r e s e v e r a l w a ys in w h ic h th e r e su l t s o f th e c a lc u la t io n s m a y b e d is p la y e d . T h e

e a s ie s t w a y i s to d is p la y th e m a s a n u m e r ic a l l i s t in g ( ta b le 5 -1 ) . I t i s o f te n d i f f ic u l t , h o w e v e r ,

to se e th e r e la t io n s h ip s b e tw e e n n u m b e r s i n a ta b le o f d a ta , s o a b e t te r m e t h o d is to d i sp la y th e

re su l t s a s a g ra p h o f th e M v a lu e s v e rs u s th e p i tc h a n g le s o f t h e s l i p v e c to r s ( f ig u r e 5 -7 ) .

A l te r n a t ive ly , the sa m e s l ip ve c to rs m a y b e p lo t ted in a lo w e r -h e m isp h e re s te re o g ra p h ic

p ro je c t io n ( f igu re 5 -8 ) . U s in g d i f fe re n t sym b o ls fo r s l ip v e c to rs o n p lan e s w i th a sh e a r s t re ss to

n o r m a l s t r e ss r a t io a b o v e o r b e lo w a c e r ta in c h o s e n v a lu e m a y in d ic a te w h ic h s l ip d i re c t io n s

w o u ld h a v e a h ig h e r c h a n c e o f e x p e r ie n c in g s l ip s in c e th is r a t io i s d i r e c t ly p r o p o r t io n a l to th e

c o e f f ic ien t o f in te rn a l f r i c t io n (C o u lo m b , 1 7 7 6 ; H a n d in , 1 9 6 9 ) .

T h e s l ip ve c to r c a lcu la t io n p ro g ra m w i l l d i sp lay , o n the c o m p u te r 's sc re e n , th e re su l t s

a s a ta b le o f d a ta , a g ra p h o f th e M va lu e s v e r sus the p i tch an g le s o f the s l ip ve c to r s , o r a s a

lo w er -h e m isp h e re s te re o gra p h ic p ro j e c t io n . T h e d a ta m a y a lso b e w r i t te n to a n A S C I I f i le o r b e

u s e d to c r e a te A uto C A D -c o m p atib le d ra wing in te rc ha n g e f i le s (D X F f i le s ) fo r p lo t t in g th e g r a p h s

a n d s te re o g ra p h ic p ro je c t io n s v ia A u to C A D .

5 .6 C r ea t in g F a u lt P o p u la t io n s

T h e s l ip ve c to r c a lc u la t io n p ro g ra m w as u se d to ge ne ra te a r t i f ic ia l fa u l t p o p u la t io n s fo r

te s t ing p a leo s t r e s s an a lys is p ro gra m s . T he se pop u la tions were c re a t ed us ing the fo l lo w ing s tep s :

1 . D e c id ing up o n the typ e o f fa u l t p o p u la t io n to t es t ( i . e . c o n ju g a te fau l t s , o r th o r h o m b ic

sym m e try fau l t s , r a n d o m ly o r ie n ted fau l t s ) .

2 . D e c id ing up o n the nu m b e r o f fa u l t s to t es t . T o o few o r to o m a n y fau l t s wi l l ad v e rs ly

a f fec t the p a l eo s t r e s s an a lys is .

3 . D e c id in g u p o n th e typ e o f s t r e ss f ie ld in wh ic h to s i tu a te th e fa u l t p o p u la t io n . T h e

F ig u r e 5 - 6 - S im p l i f i ed f lo w c h a r t d e m o n s t ra t ing the m a the m a t ica l a lg o r i thm us e d b y the s l ip

v e c to r c a lc u la t io n p ro gra m .

s nS i g m a 1 S i g m a 2 S i g m a 3 P h i P i t c h F /F

1 .0 0 -1 .0 0 -1 .0 0 0 .0 0 5 8 .4 3 ° 0 . 6 9

1 . 0 0 - 0 . 9 0 - 1 . 0 0 0 . 0 5 5 6 . 6 4 ° 0 . 7 3

1 . 0 0 - 0 . 8 0 - 1 . 0 0 0 . 1 0 5 4 . 6 7 ° 0 . 7 9

1 . 0 0 - 0 . 7 0 - 1 . 0 0 0 . 1 5 5 2 . 4 8 ° 0 . 8 7

1 . 0 0 - 0 . 6 0 - 1 . 0 0 0 . 2 0 5 0 . 0 4 ° 0 . 9 8

1 . 0 0 - 0 . 5 0 - 1 . 0 0 0 . 2 5 4 7 . 3 4 ° 1 . 1 4

1 . 0 0 - 0 . 4 0 - 1 . 0 0 0 . 3 0 4 4 . 3 2 ° 1 . 4 0

1 . 0 0 - 0 . 3 0 - 1 . 0 0 0 . 3 5 4 0 . 9 6 ° 1 . 9 0

1 . 0 0 - 0 . 2 0 - 1 . 0 0 0 . 4 0 3 7 . 2 2 ° 3 . 1 5

1 . 0 0 - 0 . 1 0 - 1 . 0 0 0 . 4 5 3 3 . 0 7 ° 1 1 . 8 6

1 . 0 0 0 . 0 0 - 1 . 0 0 0 . 5 0 2 8 . 4 8 ° 5 . 7 6

1 . 0 0 0 . 1 0 - 1 . 0 0 0 . 5 5 2 3 . 4 6 ° 2 . 2 0

1 . 0 0 0 . 2 0 - 1 . 0 0 0 . 6 0 1 8 . 0 3 ° 1 . 3 2

1 . 0 0 0 . 3 0 - 1 . 0 0 0 . 6 5 1 2 . 2 4 ° 0 . 9 4

1 . 0 0 0 . 4 0 - 1 . 0 0 0 . 7 0 6 . 1 9 ° 0 . 7 2

1 . 0 0 0 . 5 0 - 1 . 0 0 0 . 7 5 0 . 0 0 ° 0 . 5 0

1 . 0 0 0 . 6 0 - 1 . 0 0 0 . 8 0 - 6 . 1 9 ° 0 . 5 1

1 . 0 0 0 . 7 0 - 1 . 0 0 0 . 8 5 - 1 2 . 2 4 ° 0 . 4 5

1 . 0 0 0 . 8 0 - 1 . 0 0 0 . 9 0 - 1 8 . 0 3 ° 0 . 4 1

1 . 0 0 0 . 9 0 - 1 . 0 0 0 . 9 5 - 2 3 . 4 6 ° 0 . 3 8

1 . 0 0 1 . 0 0 - 1 . 0 0 1 . 0 0 - 2 8 . 4 8 ° 0 . 3 6

T a b l e 5 - 1 - T a b le o f d a ta g e ne ra te d b y th e c a lc u la t io n o f 2 1 s l ip v e c to r s r e p r ese n t in g a M v a lu e

1r a n g in g f ro m 0 .0 to 1 .0 o n a fa u l t p la n e w i th a n o r m a l v e c to r o r ie n te d a t 7 0 /0 3 0 d e g re e s . T h e F ,

2 3F , a n d F v a lu e s a re in a n y a r b i t r a r y s t r e ss u n i t s , th e p i tc h i s d e f in e d a s th e a n g le b e tw e e n th e

s t r ik e o f the fau l t p lan e a n d the s l ip v e c to r in d e g re e s , a n d the sh e a r s t re ss to no rm a l s t re ss r a t io

s n(F / F ) i s d im e n s io n le ss ( t ab le m o d i f ie d f ro m S c h im m ric h , 1 9 9 0 ) .

F ig u r e 5 - 7 - G ra p h o f 2 1 M v a lu e s fo r e a c h s l ip v e c to r a s th e y r a n g e fr o m 0 . 0 to 1 . 0 v e r s u s th e

p i tc h o f th e s l ip v e c to r s f ro m th e s t r ik e o f th e fa u l t p la ne w ith a no rm a l ve c to r o r ie n te d a t 7 0 /0 3 0

d e g re e s . T he la rg e c i rc le s re p re se n t s l ip v e c to rs w i th a sh ea r s t re ss to no rm a l s t re ss r a t io o f 0 .6

o r g re a te r a n d the sm a l l c i rc le s re p re se n t s l ip v e c to rs w i th a sh e a r s t re ss to no rm a l s t re ss r a t io

o f le ss th a n 0 .6 .

F ig u r e 5 - 8 - L o w e r -h e m isp h e re s te re o g ra p h ic p r o j ec t io n o f 2 1 s l ip v e c to r s r e p r ese n t in g v a lu e s

r a n g in g f r o m 0 . 0 to 1 . 0 o n a fa u l t p l a n e w i th a n o r m a l v e c to r o r ie n te d a t 7 0 / 0 3 0 d e g r e e s . T h e

la r g e c i r c le s re p r e se n t s l ip v e c to r s w i th a sh e a r s t r e ss to n o r m a l s t r e ss r a t io o f 0 . 6 o r g re a te r a n d

th e sm a ll c i r c le s r e p re se n t s l ip ve c to r s w ith a sh e a r s t r e ss to n o rm a l s t r e ss r a t io o f le ss th a n 0 .6 .

1 2 3o r ie n ta t io n s o f F , F , a n d F r e la t iv e to th e n o r th , u p , a n d e a s t c o o r d i n a t e a x e s a n d

th e ir r a t io M m u s t b e d e te rm ine d .

04 . D e c id in g u p o n v a lu e s to u se fo r th e c o e f f ic ie n t o f f r ic t io n (: ) a n d c o h e s io n ( C ) .

5 . D e te r m in in g th e p lu n g e a n d t r e n d o f th e n o r m a l v e c to r fo r e a c h fa u l t p l a n e w i th in th e

p o p u la t io n w h ich w i l l b e t es ted .

6 . U s in g th e s l ip v e c to r c a lc u la t io n p r o g r a m to d e te r m in e th e p i tc h o f th e s l ip v e c to r a n d

th e sh e a r s t r e ss to no rm a l s t r e ss r a t io fo r e ac h fa u l t p la ne a t th e d e c id e d u p o n va lu e o f

7 . D e c id ing up o n a sh e a r s t re ss to no rm a l s t re ss r a t io c u to f f v a lue to u se s inc e fau l t s wi th

a s u ff ic ie n t ly lo w r a t io w i l l n o t e x p e r ie n c e s l ip u n d e r re a l i s t ic g e o lo g ic c o n d i t io n s a n d

to ss in g o u t th o se fau l t s wh ich ha v e a ra t io b e lo w tha t cu to f f .

8 . C r e a t in g d a ta f i le s o f th e fa u l t 's o r ie n ta t io n s a n d s l ip d i r e c t io n s in th e p ro p e r fo rm a t fo r

e n t ry in to the va r io u s p a leo s t re ss a n a lys i s p ro g ra m s .

T h e a r t i f ic ia l fa u l t p o p u la t io n s g en e ra te d in th is m a n ne r w e r e th e n ru n th ro u g h se v e ra l

p a l e o s t r e s s a n a lys is p ro gra m s to d e te rm in e i f th e c a lc u la te d p a le o s t r e s s te n so rs c o r re sp o n d e d

to th e in i t ia l s t r e ss f ie ld s u s e d to c r e a te th e p o p u la t io n s . T h is p r o v i d e s a n in d e p e n d e n t m e a n s

o f a s se s s in g t h e a c c u r a c y o f c o m p u ta t io n a l m e th o d s o f p a lo s tr e ss a n a lys is s in c e th e p o p u la t io n s

w e re c re a ted us ing the sa m e in i t i a l m a the m a t ica l as su m p t i o n s a s th o se us e d b y the a n a lys i s

p ro g ra m s .

C H A P T E R 6

A N G E L IE R 'S M E T H O D O F P A L E O S T R E S S A N A L Y S IS

In 1 9 7 5 , J a c q u e s A n g e l i e r o f th e U n iv e rs i t é P ie r r e e t M a r i e C u r i e in P a r i s p ro p o se d

a n e w c o m p u t a t i o n a l m e th o d fo r p a l e o s t r e s s a n a ly s i s w h ic h h e s u b s e q u e n t ly m o d i f i e d o v e r

t i m e (A n g e l i e r , 1 9 7 5 ; A n g e l i e r , 1 9 7 9 ; A n g e l i e r , e t . a l . , 1 9 8 2 ; A n g e l i e r , 1 9 8 4 ; A n g e l i e r ,

1 9 8 9 ) . A n g e l i e r ' s m e th o d a t t e m p ts to i t e ra t i v e ly d e te rm in e a p a le o s t r e s s t e n so r fo r a g iv e n

f a u l t p o p u l a t i o n s u c h t h a t t h e a n g u l a r d i v e r g e n c e b e t w e e n t h e o b s e r v e d s t r ia t i o n s i n t h e f a u l t

p la n e s a n d th e p re d ic te d s l i p d i r e c t i o n s a re m in im iz e d (A n g e l i e r , e t . a l . , 1 9 8 2 ) .

I o b ta in e d a c o m p i l e d v e r s io n o f A n g e l i e r ' s p ro g ra m f ro m C h r i s t o p h e r B a r to n o f th e

L a m o n t -D o h e r ty G e o lo g ic a l O b se rv a to ry in Ju n e , 1 9 9 0 . T h e p ro g ra m w a s w r i t t e n b y A n g e l i e r

in F O R T R A N fo r a n IB M P C o r c o m p a t i b le c o m p u te r w i t h a n 8 0 2 8 7 m a th c o p ro c e s s o r .

I d e t e rm in e d th a t t h i s p ro g ra m w a s o pe ra t i n g c o r re c t ly b y e x a m in in g s e v e ra l p u b l i s h e d

fa u l t p o p u la t i o n s fo r w h ic h A n g e l i e r ' s m e th o d p a l e o s t r e s s t e n s o rs w e re g iv e n a n d c o m p a r in g

m y re s u l t s to th e p u b l i s h e d o n e s (A n g e l i e r , e t . a l . , 1 9 8 2 ; A n g e l i e r , 1 9 8 4 ) . T h i s d o n e , I b e g a n

to e v a lu a te th e p e r f o rm a n c e o f A n g e l i e r ' s m e th o d u s in g a r t i f i c ia l f a u l t p o p u la t i o n s .

6 .1 P r o g r a m A ssu m p t io n s

A n g e l i e r ' s m e t h o d o f p a l e o s t r e s s a n a l y s i s i s b a s e d u p o n t h e f o l l o w i n g t w o v e r y

im p o r t a n t i n i t i a l a s s u m p t i o n s (A n g e l i e r , 1 9 8 9 ) .

1 . A l l fa u l t s w h ic h m o v e d d u r in g a s in g l e t e c t o n i c e v e n t m o v e d in d e p e n d e n t ly o f o n e

a n o th e r a n d in a m a n n e r c o n s i s te n t w i t h a u n iq u e , s t a t i c s t r e s s t e n s o r .

2 . F a u l t s a r e a s su m e d to s l i p o n p re - e x i s t i n g p la n a r d i s c o n t in u i t i e s in th e d i r e c t i o n o f th e

m a x i m u m r e so lv e d s h e a r s t r e s s w i th in th e f a u l t p la n e ( i . e . a t r i g h t a n g le s to th e d i r e c t i o n o f

z e ro s h e a r s t r e s s ) .

6 .2 T h e o r y

T h e m a th e m a t i c s in th i s s e c t i o n ro u g h ly fo l l o w th e d e r i v a t i o n s g iv e n in A n g e l i e r , e t .

a l . (1 9 8 2 ) .

A l lo w F t o b e t h e u n k n o w n r e g i o n a l s t r e s s t e n s o r a c t i n g u p o n a f a u l t p l a n e w i t h a u n i t

n o r m a l v e c t o r N a n d a u n i t s l i p v e c t o r S ( f i g u r e 6 - 1 ) . T h e s t r e s s v e c t o r T a c t i n g u p o n t h e f a u l t

p l a n e m a y b e d e f in e d b y

T = F @ N (1 )

w h e re (F @ N ) i s t h e i n n e r , o r d o t , p r o d u c t o f t e n s o r F a n d v e c t o r N (M a rs d e n a n d T ro m b a ,

1 9 8 1 , p . 2 0 ) a n d t h e c o m p o n e n t s o f T o n N a n d S a re

N @ T = N @ F @ N (2 )

S @ T = S @ F @ N (3 )

re s p e c t i v e ly .

S i n c e th e d i r e c t i o n o f th e s t r i a t i o n s i n th e f a u l t p l a n e i s t a k e n t o b e th e m a x im u m

F ig u r e 6 - 1 - G e o m e t r y o f th e s t r e s s e s o n a s t r ia t e d f a u l t p l a n e w i t h a u n i t n o r m a l v e c t o r (N ) ,

a u n i t s l i p v e c t o r (S ) , a n d a s t r e s s v e c t o r (F @ N ) a c t i n g u p o n i t ( f i g u re m o d i f i e d f ro m

A n g e l i e r , e t . a l . , 1 9 8 2 ) .

re s o lv e d s h e a r s t r e s s d i r e c t i o n , t h e fo l l o w in g tw o e q u a t io n s m a y b e w r i t t e n

F @ N = (N @ F @ N ) N + (S @ F @ N ) S (4 )

S @ F @ N $ 0 (5 )

E q u a t io n (4 ) m a y b e s im p l i f i e d th r o u g h t h e f o l l o w in g s t e p s

(S @ F @ N ) S = F @ N - (N @ F @ N ) N (6 )

(S @ F @ N ) = 2 F @ N - (N @ F @ N ) N 2 (7 )2

(S @ F @ N ) = 2 F @ N 2 - (N @ F @ N ) (8 )2 2 2

S @ F @ N = ± [2 F @ N 2 - (N @ F @ N ) ] (9 )2 2 1/2

a n d c o m b i n i n g e q u a t io n s (5 ) a n d (9 ) y i e l d s

S @ F @ N = + [2 F @ N 2 - (N @ F @ N ) ] (1 0 )2 2 1/2

a m a th e m a t i c a l r e la t i o n s h ip w h ic h w i l l b e u s e d la te r i n th i s a n a ly s i s .

I n d e s c r i b i n g t h e o r i e n t a t i o n o f t h e f a u l t p l a n e s a n d t h e i r a s s o c i a t e d s t r i a t i o n s u s e d i n

th e p a l e o s t r e s s a n a ly s i s , t h r e e in d e p e n d e n t a n g le s a re d e f in e d i n a n o r th , u p , a n d e a s t

g e o g r a p h i c c o o r d i n a t e s y s t e m . T h e t re n d o f t h e f a u l t 's d i p d i r e c t i o n ( d ) , t h e f a u l t 's d i p a n g l e

( p ) , a n d th e p i t c h a n g le o f t h e f a u l t ' s s l i p v e c to r ( i ) . T h e p i t c h a n g le ( i ) i s d e f in e d a s t h e

c lo c k w is e a n g le ( l o o k in g d o w n th e f a u l t n o rm a l 's p lu n g e a t t h e f o o t w a l l b lo c k ) b e tw e e n t h e

s l i p v e c to r a n d th e f a u l t ' s s t r i k e d i r e c t i o n ( t h e t r e n d o f t h e p o l e t o th e f a u l t p l a n e + B / 2 ) s u c h

th a t 0 < i < B fo r a n o rm a l f a u l t a n d B < i < 2B fo r a re v e rs e fa u l t ( f i g u re 6 -2 ) .

T h e a n g l e s d , p , a n d i a r e t h o s e c o m m o n l y m e a s u re d in th e f i e l d b y g e o lo g i s t s w i th a

F ig u r e 6 - 2 - D i a g r a m o f a f a u l t p l a n e l o o k i n g d o w n t h e p l u n g e o f t h e n o r m a l v e c t o r ( N ) to

t h e f a u l t a t t h e f o o t w a l l b l o c k s h o w i n g a s l i p v e c t o r a l i g n e d a t 3B / 4 . I n A n g e l i e r ' s ( A n g e l i e r ,

e t . a l . , 1 9 8 2 ) n o t a t i o n a l s y s t e m , t h i s im p l i e s a n o rm a l f a u l t ( t h e s l i p v e c to r p o in t s i n t h e

d i r e c t i o n o f m o v e m e n t o f t h e l o w e r b l o c k ) .

c o m p a s s a n d c l i n o m e te r . T h e a d v a n t a g e i n u s in g th e s e a n g l e s i s t h a t t h e y a r e a l l i n d e p e n d e n t

o f o n e a n o th e r s o a n e r r o r i n m e a s u r i n g o n e a n g le w i l l n o t a f f e c t t h e o th e r t w o .

T h e th re e c o m p o n e n t s o f th e u n i t n o rm a l v e c to r to t h e f a u l t p la n e m a y th u s b e d e f in e d

in te rm s o f d , p , a n d i a s

1 N = [ s i n ( d ) s i n ( p ) ]

2N = [c o s (d ) s in (p ) ] (1 1 )

3N = c o s ( p )

a n d , s im i l a r ly , fo r th e s l i p v e c t o r c o m p o n e n t s

1S = - [ s i n ( i ) c o s ( p ) s i n ( d ) ] + [ c o s ( i ) c o s ( d ) ]

2S = - [ s in ( i ) c o s (p ) c o s (d ) ] - [ c o s ( i ) s i n (d ) ] (1 2 )

3S = [ s i n ( i ) s i n ( p ) ]

I f F i s th e u n k n o w n r e g i o n a l s t r e s s t e n s o r f o r a g i v e n f a u l t , a t e n s o r F ' m a y b e d e f in e d

s u c h t h a t

1 2F = t F ' + t I (1 3 )

1 2w h e re t a n d t a r e a n y p o s i t i v e c o n s ta n t s a n d I i s a n y i s o t ro p ic 3 x 3 te n s o r . I t c a n b e

d e m o n s t r a t e d th a t m u l t i p ly in g F ' b y a p o s i t i v e c o n s t a n t a n d a d d i n g a n i s o t r o p i c t e n s o r I t o i t

w i l l n o t c h a n g e th e s e n s e o r d i r e c t i o n o f t h e p re d ic te d s t r i a t i o n s o n th e fa u l t p la n e .

1,2 2,1 1,3S i n c e F i s a 3 x 3 s y m m e t r i c t e n s o r , i t h a s s ix d e g re e s o f f r e e d o m ( i . e . F = F , F

3,1 2,3 3,2= F , a n d F = F ) . T h e t e n s o r F ' i s t h u s t e rm e d th e r e d u c e d d e v i a to r i c s t r e s s t e n s o r

1 2(A n g e l i e r , 1 9 7 9 ; A n g e l i e r , e t . a l . , 1 9 8 4 ) a n d h a s fo u r d e g re e s o f f r e e d o m s in c e t a n d t m a y

h a v e a n y a rb i t r a ry p o s i t i v e v a lu e s . F u r th e rm o re , i t i s a lw a y s p o s s ib l e t o c h o o s e p o s i t i v e

1 2v a l u e s f o r t a n d t s u c h t h a t

1,1 2,2 3,3F + F + F = 0 (1 4 )

1,1 2,2 3,3F + F + F = (3 /2 ) (1 5 )2 2 2

F o r n u m b e r s s a t i s fy i n g e q u a t i o n s ( 1 4 ) a n d ( 1 5 ) , a u n i q u e n u m b e r R (m o d u lo 2B ) c a n

b e f o u n d s u c h t h a t

1,1F = c o s (R )

2,2F = c o s [R + ( 2B / 3 ) ] (1 6 )

3,3F = c o s [R + ( 4B / 3 ) ]

w h i c h y i e l d s t h e r e d u c e d s t r e s s t e n s o r F '

* c o s (R ) " ( *

* " c o s [R + ( 2B / 3 ) ] $ * (1 7 )

* ( $ c o s [R + ( 4B / 3 ) ] *

1,2 2,1 2,3 3,2 1,3 3,1w h e re " , $ , a n d ( a r e t h e s h e a r s t r e s s t e rm s (F , F , F , F , F , a n d F ) o f t h e t e n s o r .

A l l t e n s o r s F ' a n d F w h ic h s a t i s fy e q u a t io n (1 3 ) y i e ld th e s a m e e ig e n v e c t o r s a n d

e ig e n v a l u e s a n d th u s t h e p r in c ip a l s t r e s s a x e s o r i e n t a t i o n s a n d m a g n i tu d e s a re t h e s a m e fo r

e a c h .

0 0 0 0L e t R , " , $ , a n d ( b e v a lu e s fo r th e a p r io r i e s t i m a t e o f F ' a n d F , F , F , a n d FR " $ (

b e t h e i r s t a n d a r d d e v ia t i o n s . I f t h e r e i s n o a p r io r i e s t i m a t e o f F ' , t h e s t a n d a rd d e v i a t i o n s

0 0 0 nm u s t b e m a d e v e r y la rg e . A l s o , l e t (d , p , i ) b e d a t a a n d (F , F , F ) b e th e s ta n d a r dd p i

d e v ia t i o n s o f t h a t d a ta fo r e a c h fa u l t n .

In t h e g e n e ra l c a s e , n o t e n s o r F ' e x a c t l y s a t i s f i e s e q u a t i o n ( 1 0 ) o n e a c h f a u l t .

nT h e r e f o r e , w h a t i s n e e d e d i s a t e n s o r F ' a n d a n e w s e t o f d a ta (d , p , i ) w h ic h e x a c t ly s a t i s f i e s

th e e q u a t io n . S in c e th e re a re a n in f in i t e n u m b e r o f s u c h s o lu t io n s , t h e s o lu t io n w h ic h

m in im iz e s th e fo l l o w in g s u m (s ) fo r a p o p u la t i o n o f n fa u l t s i s u s e d .

k=1 6 n 0 k 0 k 0 k 0s = E { [ (d - d ) /F ] + [ (p - p ) /F ] + [ ( i - i ) /F ] } + [ (R -R ) /F ] + d 2 p 2 i 2 R 2

0 0 0[ (" -" ) /F ] + [ ($ -$ ) /F ] + [ (( -( ) /F ] (1 8 )" 2 $ 2 ( 2

T h is i s a n o n - l i n e a r l e a s t - s q u a re s p ro b le m w h o s e so lu t io n i s o b t a in e d b y a c o m p l i c a te d

i t e ra t i v e a lg o r i th m w h ic h m a y b e fo u n d in A n g e l i e r , e t . a l . (1 9 8 2 ) .

T h e p ro g ra m u t i l i z in g A n g e l i e r ' s m e th o d o f p a l e o s t r e s s a n a ly s i s r e q u i r e s f i v e i t e m s

o f in fo rm a t io n to ru n - - a tw o l e t t e r c o d e (d e s c r ib e d in m o re d e t a i l b e lo w ) d e s c r ib in g th e t y p e

o f s t ru c tu r e w h i c h w i l l b e a n a l y z e d , th e f a u l t ' s s t r i k e (0 ° 6 3 6 0 ° ) , t h e f a u l t ' s d i p a n g le ( 0 ° 6

9 0 ° ) , t h e f a u l t ' s d ip d i r e c t i o n (N , E , S , o r W fo r n o r th , e a s t , s o u th , o r w e s t r e s p e c t iv e ly ) , a n d

th e t r e n d ( 0 ° 6 3 6 0 ° ) o f t h e s t r i a t i o n s o n th e fa u l t s u r f a c e .

T h e tw o le t t e r c o d e u se d b y A n g e l i e r i s t o a l l o w a w id e v a r i e ty o f g e o lo g ic s t ru c tu re s

s u c h a s fa u l t s , j o i n t s , t e n s io n g a s h e s , d ik e s , b e d d i n g p l a n e s , m y l o n i t i c fo l i a t io n s , c l e a v a g e s ,

m in e ra l l i n e a t io n s , fo l d s , e t c . t o b e s u b je c t e d to p a l e o s t r e s s a n a ly s i s . T h e c o d e s u se d in m y

a n a ly se s a re C N , C I , C D , a n d C S f o r , r e s p e c t i v e l y , s t r i a t e d n o r m a l , r e v e r s e ( in v e r s e ) , d e x t r a l ,

o r s in i s t r a l f a u l t s w i t h a k n o w n s e n s e o f s h e a r .

T h i s i n f o r m a t i o n m u s t b e w r i t t e n t o a n A S C I I d a t a f i l e i n a s p e c i a l fo r m a t s o t h a t i t

m a y b e c o r re c t ly re a d b y A n g e l i e r ' s p ro g r a m . A d a t a f i l e c r e a t io n p ro g ra m e x i s t s w h ic h

c re a t e s th e s e fo rm a t te d f i l e s w h e n th e i n i t i a l d a t a i s i n t e r a c t iv e ly e n t e r e d . E a c h f a u l t d a tu m

m u s t b e e n t e re d a s a s in g l e l i n e o f c h a ra c te r s a n d in te g e r s s e p a ra te d b y a s in g l e b l a n k s p a c e .

A s a n e x a m p l e , a c o n j u g a t e s e t o f t w o n o r m a l fa u l t s w i t h a n e a s t - w e s t s t r ik e a n d a d i p

o f 4 5 ° ( f i g u re 6 -3 ) w o u ld b e e n t e r e d in t o th e d a t a f i l e c r e a t io n p r o g r a m a s

C N 0 9 0 4 5 S 1 8 0

C N 2 7 0 4 5 N 0 0 0

s in c e t h e f i r s t f a u l t i s n o rm a l , h a s a s t r i k e o f 0 9 0 ° , i s d ip p in g 4 5 ° to th e s o u th , a n d h a s

s t r i a t i o n s w i t h a t r e n d o f 1 8 0 ° . T h e s e c o n d f a u l t i s a l s o n o r m a l , h a s a s t r ik e o f 2 7 0 ° , i s

d ip p in g 4 5 ° to t h e n o r th a n d h a s s t r i a t i o n s w i th a t r e n d o f 0 0 0 ° . T h e d a ta f i l e c r e a t io n p ro g ra m

a l s o a s k s fo r th e m a g n e t i c d e v i a t i o n o f th e m e a s u re m e n t s i n d e g re e s , a v a lu e fo r th e

in s t ru m e n t e r ro r in d e g re e s , t h e a u th o r 's n a m e , th e s i t e n a m e , t h e d a te , a n d c o m m e n t s a b o u t

th e g e o lo g y o f t h e s i t e .

O n c e t h e p ro p e r ly - fo rm a t t e d A S C I I d a ta f i l e h a s b e e n c re a t e d a n d r e a d i n to th e

p ro g ra m , th e c a lc u la t i o n s a re p e r f o rm e d .

0 0 0W h e n a l l o f t h e d a ta h a s b e e n e n t e re d in to t h e p ro g ra m , t h e t h re e a n g le s d , p , a n d i

a r e d e t e r m in e d f o r e a c h fa u l t i n t h e p o p u l a t io n . T h e s t a n d a rd d e v i a t io n s F , F , a n d F o fd p i

th e s e a n g le s a re a l s o d e te rm in e d f ro m th e e n te re d m e a s u re m e n t e r r o r t e rm .

T h e n e x t s t e p i n t h e p ro g ra m i s to s e t t h e a p r io r i c o n s t r a i n t s f o r t h e f o u r p a r a m e t e r s

0 0 0 0(R , " , $ , a n d ( ) d e s c r i b i n g t h e r e d u c e d s t r e s s t e n s o r F ' ( e q u a t io n 1 7 ) to z e ro a n d to s e t t h e

a p r io r i s t a n d a r d d e v ia t i o n s t o 4B f o r F a n d t o 1 0 0 f o r F , F , a n d F . R " $ (

A n i t e ra t i v e a lg o r i th m (A n g e l i e r , e t . a l . , 1 9 8 2 ) i s t h e n u s e d t o d e t e r m i n e v a l u e s f o r [R ,

1 n" , $ , ( , (d , p , i ) , . . . (d , p , i ) ] s u c h th a t e q u a t i o n (1 8 ) i s m in im iz e d .

O n c e a r e d u c e d s t r e s s t e n s o r F ' h a s b e e n c a l c u la t e d , t h e t h r e e e ig e n v a l u e s a n d

e i g e n v e c t o r s o f F ' m a y b e d e t e rm in e d u s in g s t a n d a r d l i n e a r a lg e b ra t e c h n i q u e s (A n to n , 1 9 8 1 ,

p . 2 6 1 -2 8 4 ; F rö b e rg , 1 9 8 5 , p . 2 2 -2 6 ) . T h e s e e ig e n v a l u e s a n d e ig e n v e c t o rs c o r re s p o n d to th e

1 2 3m a g n i tu d e s a n d o r ie n t a t io n s o f th e th r e e p r in c ip a l s t r e s s a x e s F , F , a n d F r e s p e c t i v e ly .

T h e p ro g ra m re s u l t s a r e d i s p la y e d o n th e c o m p u te r 's s c re e n w h e n th e c a l c u l a t i o n s h a v e

f in i s h e d a n d th e u se r h a s th e o p t io n o f s a v in g th e m to a n A S C II d a t a f i l e o r p e r fo rm in g a n o th e r

a n a ly s i s .

1T h e o u t p u t d a t a c o n s i s t s o f th e p l u n g e s a n d t r e n d s o f th e th r e e p r in c ip a l s t r e s s a x e s F ,

2 3 2 3 1 3F , a n d F a n d t h e i r r e la t i v e m a g n i tu d e s . T h e s t r e s s r a t i o M , d e f i n e d a s [ (F - F ) / (F - F ) ] ,

i s a l s o d i s p la y e d .

T h e d a t a f o r e a c h p a l e o s t r e s s a n a l y s i s I o b t a i n e d w a s p l o t t e d o n l o w e r -h e m i s p h e r e

F ig u r e 6 - 3 - L o w e r -h e m i s p h e re s t e r e o g ra p h i c p ro j e c t io n s h o w in g a c o n ju g a t e s e t o f tw o

n o rm a l f a u l t s w i th a n e a s t -w e s t s t r i k e a n d a d ip o f 4 5 °

s te re o g ra p h ic p ro je c t io n s b y a T u rb o P a s c a l v e r s io n 3 .0 1 p ro g ra m I w ro te fo r th a t p u rp o s e

(A p p e n d i x D ) . T h e p r o g r a m re a d th e f a u l t p o p u la t io n d a t a a n d th e p r e d i c t e d p r in c ip a l s t r e s s

a x e s o r i e n ta t i o n s a n d c re a te d A u to C A D s c r i p t f i l e s fo r p lo t t in g s te re o n e ts v ia A u to C A D .

A n g e l i e r 's m e t h o d o f p a l e o s t r e s s a n a l y s i s w a s c h o s e n fo r t e s t in g fo r s e v e r a l re a s o n s .

I h a d a w o rk in g c o p y o f th e p ro g ra m fo r a n IB M P C o r c o m p a t ib le c o m p u te r , t h e p ro g ra m

p e r fo rm e d c a l c u l a t i o n s fo r r e a s o n a b ly - s iz e d f a u l t p o p u l a t i o n s i n r e la t i v e ly s h o r t a m o u n t s o f

t i m e , A n g e l i e r ' s m e th o d o f p a l e o s t r e s s a n a ly s i s i s w id e ly -u s e d a n d i s t h e s t a n d a r d b y w h i c h

m o s t o th e r s a r e ju d g e d , a n d th e m e th o d h a s b e e n u s e d in s e v e ra l p u b l i s h e d f i e ld s t u d i e s

(A n g e l i e r , 1 9 8 4 ; A n g e l i e r , e t . a l . , 1 9 8 5 ; A n g e l i e r , 1 9 9 0 ) .

C H A P T E R 7

R E C H E S ' M E T H O D O F P A L E O S T R E S S A N A L Y S IS

In 1 9 8 7 , Z e 'e v R e c h e s o f th e H e b re w U n iv e rs i ty in J e ru sa le m p ro p o se d w h a t h e c la im e d

to b e a n e w a n d i m p ro v e d m e th o d o f c o m p u ta t io n a l p a l e o s t r e s s a n a ly s i s (R e c h e s , 1 9 8 7 ) . T h e

im p r o v e m e n t o v e r p re v io u s m e th o d s w a s th e i n c o rp o ra t io n o f th e C o u lo m b fa i l u re c r i t e r io n

in to th e c a l c u l a t i o n s . T h i s a l l o w s th e c o h e s io n a n d c o e f f i c i e n t o f f r i c t i o n to b e c o n s t r a in e d fo r

th e f a u l t p o p u la t i o n s a n d u s e s a l i n e a r in v e rs io n to c a lc u la t e a s t r e s s t e n s o r r a th e r th a n

A n g e l i e r ' s m u c h m o re c o m p l i c a te d n o n - l in e a r o n e .

I o b ta in e d a c o m p i l e d v e r s i o n o f R e c h e s ' p ro g ra m f ro m K e n n e t h H a rd c a s t l e o f th e

U n iv e r s i ty o f M a s sa c h u s e t t s a t A m h e r s t i n A u g u s t , 1 9 8 9 . T h e p ro g ra m w a s w r i t t e n b y

H a rd c a s t l e in M ic ro so f t B A S IC v e r s io n 5 .6 0 fo r a n IB M P C o r c o m p a t ib le c o m p u t e r w i th a n

8 0 2 8 7 m a th c o p ro c e s s o r . U s in g th i s p ro g ra m , H a rd c a s t l e p e r fo rm e d p a l e o s t r e s s a n a l y s e s o n

fa u l t d a ta f ro m e a s te rn V e rm o n t a n d w e s te rn N e w H a m p s h i r e (H a rd c a s t l e , 1 9 8 9 ) .

I d e t e rm in e d th a t t h i s p ro g ra m w a s o pe ra t i n g c o r re c t ly b y e x a m in in g s e v e ra l p u b l i s h e d

fa u l t p o p u la t i o n s fo r w h ic h R e c h e s ' m e th o d p a l e o s t r e s s t e n s o r s w e re g iv e n a n d c o m p a r in g m y

r e s u l t s to t h e p u b l i s h e d o n e s ( A n g e l i e r , 1 9 8 4 ; R e c h e s , 1 9 8 7 ) . T h i s d o n e , I b e g a n t o e v a l u a t e

th e p e r f o rm a n c e o f R e c h e s ' m e th o d u s in g a r t i f i c ia l f a u l t p o p u la t i o n s .

7 .1 P r o g r a m A ssu m p t io n s

R e c h e s ' m e th o d o f p a l e o s t r e s s a n a l y s i s i s b a s e d u p o n th e fo l l o w in g th re e a s s u m p t io n s

(R e c h e s , 1 9 8 7 ) .

1 . F a u l t s a r e a s s u m e d t o s l i p i n t h e d i r e c t i o n o f t h e m a x i m u m r e s o l v e d s h e a r s t re s s w i t h i n

th e f a u l t p la n e ( i . e . a t r ig h t a n g le s to th e d i r e c t i o n o f z e ro s h e a r s t r e s s ) .

2 . T h e m a g n i tu d e s o f t h e s h e a r a n d n o rm a l s t r e s se s a c t i n g u p o n a f a u l t p l a n e s a t i s fy th e

s 0 nC o u lo m b fa i l u re c r i t e r io n (F > C + : F ) . T h i s a s s u m p t i o n i m p l i e s t h a t f a u l t s w i l l

s l i p u n d e r g e o lo g ic a l l y - r e a l i s t i c c o n d i t io n s .

3 . S l ip o n a fa u l t o c c u r s w i th in a r e l a t iv e ly s t a t i c s t r e s s f i e l d a n d th e c o e f f i c i e n t o f

0f r i c t i o n : a n d c o h e s i o n C t e r m s f o r a f a u l t m a y b e re p re s e n t e d b y t h e i r m e a n v a lu e s .

7 .2 T h e o r y

T h e m a t h e m a t i c s i n t h i s s e c t i o n r o u g h l y f o l l o w s t h e d e r iv a t io n s g iv e n in R e c h e s (1 9 8 7 ) .

F o r e a c h f a u l t p l a n e i n t h e p o p u l a t i o n t o t e s t , t h e f o l l o w i n g i t e m s o f in f o r m a t i o n a r e

k n o w n - - t h e o r i e n t a t i o n o f th e n o rm a l v e c to r to th e f a u l t p l a n e , t h e o r i e n t a t i o n o f th e f a u l t ' s

1s l i p v e c to r , a n d th e f a u l t ' s s e n se o f s l i p . A s su m in g a g e o g ra p h i c c o o r d i n a t e s y s t e m w i th X

2 3n o r th w a rd , X e a s tw a rd , a n d X d o w n w a rd , t h e t w o u n i t v e c to r s r e p re s e n t in g th e n o rm a l a n d

s l ip d i r e c t i o n s m a y b e r e p re s e n te d b y

1 2 3< N , N , N > (1 )

1 2 3< S , S , S > (2 )

i iw h e re N a n d S a re t h e d i r e c t i o n c o s i n e s f o r t h e n o rm a l a n d s l i p v e c to r s a n d th e 1 , 2 , a n d 3

s u b s c r i p t s re fe r t o th e n o r t h , e a s t , a n d d o w n d i r e c t i o n s re s p e c t i v e ly .

T h e d i re c t i o n c o s in e s in e q u a t io n s (1 ) a n d (2 ) s a t i s fy t h e f o l l o w in g id e n t i t i e s :

1 2 3N + N + N = 1 (3 )2 2 2

1 2 3S + S + S = 1 (4 )2 2 2

1 1 2 2 3 3N S + N S + N S = 0 (5 )

i iG iv e n th a t S r e p re s e n t s t h e c o m p o n e n t s o f th e f a u l t ' s s l i p v e c to r , N re p re s e n t s t h e

ic o m p o n e n t s o f th e f a u l t ' s n o rm a l v e c to r , a n d B re p re s e n t s t h e c o m p o n e n t s o f th e v e c t o r

o r th o g o n a l to b o th S a n d N ( i . e . B i s t h e n o rm a l v e c to r to th e m o v e m e n t p l a n e ) , t h e n

B = N x S (6 )

w h e r e t h e s y m b o l x r e p r e s e n t s t h e t h e o p e r a t i o n o f d e t e r m i n i n g t h e c r o s s p r o d u c t o f t w o v e c t o r s

(M a rs d e n a n d T ro m b a , 1 9 8 1 , p . 2 5 ) .

A s s u m p t i o n 1 i n s e c t i o n 7 . 1 s t a t e s t h a t th e r e s o l v e d s h e a r s t re s s p a r a l l e l to v e c t o r B ( i . e .

a t r i g h t a n g l e s t o v e c t o r S ) i s e q u a l to z e ro . R e s o lv in g th e s t r e s se s p a ra l l e l t o B ( J a e g e r a n d

i,jC o o k , 1 9 7 9 , p . 1 7 - 2 4 ) , d e n o t in g n in e c o m p o n e n t s o f t h e s t r e s s t e n s o r a s F (w h e re i = 1 , 2 , o r

3 a n d j = 1 , 2 , o r 3 ) , a n d u s in g th e i d e n t i t i e s in e q u a t io n s (3 ) , (4 ) , a n d (5 ) a l l o w s o n e to s e t u p

th e fo l l o w in g re l a t i o n s h ip :

1 1 1,1 3,3 2 2 2,2 3,3 2 3 2 3 2,3N B (F - F ) + N B (F - F ) + ( N B + B N ) F +

1 3 1 3 1,3 1 2 1 2 1,2(N B + B N ) F + (N B + B N ) F = 0 (7 )

I n a s i m i l a r m a n n e r , t h e s t r e s s e s m a y b e r e s o l v e d p a r a l l e l to t h e v e c t o r N (w h ic h i s t h e

nf a u l t ' s n o r m a l s t r e s s F )

2,2 3,3 2,3 2,2 Equation (3) in Reches' (1987) paper is incorrect with : , : , and : representing what should be F ,1

3,3 2,3F , and J respectively. This error has been corrected for equation (11).

n 1 1,1 3,3 2 2,2 3,3 3,3 2 3 2,3F = [ N (F - F ) + N (F - F ) + F + N N 2F +2 2

1 3 1,3 1 3 1,3 1 2 1,2 N N 2F + N N 2F + N N 2F ] (8 )

sa n d t h e v e c t o r S ( t h e f a u l t ' s s h e a r s t r e s s F ) .

s 1 1 1,1 3,3 2 2 2,2 3,3 2 3 2 3 2,3F = N S (F - F ) + N S (: - : ) + ( N S + S N ) : +

1 3 1 3 1,3 1 2 1 2 1,2(N S + S N ) F + (N S + S N ) F (9 )

A s s u m p t i o n 2 i n s e c t i o n 7 .1 s t a t e s t h a t a f a u l t m u s t s a t i s fy t h e C o u l o m b f a i l u r e

c r i t e r i o n .

s 0 nF = C + : F (1 0 )

S u b s t i t u t i n g e q u a t io n s (8 ) a n d (9 ) in to e q u a t io n (1 0 ) y i e ld s 1

1 1 1,1 3,3 2 2 2,2 3,3 2 3 2 3 2,3N S (F - F ) + N S (F - F ) + ( N S + S N ) F +

1 3 1 3 1,3 1 2 1 2 1,2(N S + S N ) F + (N S + S N ) F =

0 1 1,1 3,3 2 2,2 3,3 3,3 2 3 2,3C + : [ N (F - F ) + N (F - F ) + F + N N 2F + 2 2

1 3 1,3 1 3 1,3 1 2 1,2N N 2F + N N 2F + N N 2F ] (1 1 )

E q u a t io n (1 1 ) m a y b e re w r i t t e n a s

1 1 1 1,1 3,3 2 2 2 2,2 3,3[ (N S - :N ) (F - F ) ] + [ (N S - :N ) (F - F ) ] + 2 2

2 3 2 3 2 3 2,3 1 3 1 3 1 3 1,3[ (N S + S N - 2:N N ) F ] + [ (N S + S N - 2:N N ) F ] +

1 2 1 2 1 2 1,2 0 3,3[ (N S + S N - 2:N N ) F ] = C + :F (1 2 )

E q u a t io n s (7 ) a n d ( 1 2 ) m a y b e w r i t t e n fo r e a c h f a u l t i n a p o p u la t i o n o f n fa u l t s r e s u l t i n g

in a t o ta l o f 2 n e q u a t io n s . T h u s , fo r a g iv e n f a u l t p o p u la t i o n , t h e l e f t s i d e s o f th e tw o e q u a t io n s

0 3,3( i . e . t h e s id e s w h ic h a re e q u a l to 0 a n d C + :F r e s p e c t iv e ly ) m a y b e u s e d to c r e a t e a 2 n x 5

m a t r ix o f th e fo rm

1 1 1 2 2 1 2 3 2 3 1 1 3 1 3 1 1 2 1 2 1* [N B ] [N B ] [(N B + B N )] [(N B + B N )] [(N B + B N )] *

* ... ... ... ... ... *

1 1 n 2 2 n 2 3 2 3 n 1 3 1 3 n 1 2 1 2 n* [N B ] [N B ] [(N B + B N )] [(N B + B N )] [(N B + B N )] *

1 1 1 1 2 2 2 1 2 3 2 3 2 3 1 1 3 1 3 1 3 1 1 2 1 2 1 2 1* [N S -:N ] [N S -:N ] [N S +S N -2:N N ] [N S +S N -2:N N ] [N S +S N -2:N N ] *2 2

* ... ... ... ... ... *

1 1 1 n 2 2 2 n 2 3 2 3 2 3 n 1 3 1 3 1 3 n 1 2 1 2 1 2 n* [N S -:N ] [N S -:N ] [N S +S N -2:N N ] [N S +S N -2:N N ] [N S +S N -2:N N ] *2 2

w h e re t h e s u b s c r ip t s 1 th r o u g h n i n d i c a t e th e p a ra m e t e r s a s s o c i a t e d w i th e a c h o f th e n f a u l t s .

D e n o t in g th e m a t r ix i n e q u a t io n (1 3 ) a s A , t h e fo l l o w in g re l a t i o n s h ip m a y b e s e t u p

A x D = F (1 4 )

w h e re D i s th e v e c t o r

1,1 3,3 2,2 3,3 2,3 1,3 1,2< (F - F ) , (F - F ) , F , F , F > (1 5 )

o f th e u n k n o w n s t r e s se s to b e s o lv e d fo r a n d F i s th e v e c t o r

1 n 0 3,3 1 0 3,3 n< [0 ] , . . . , [0 ] , [C + : F ] , . . . , [C + : F ] > (1 6 )

0 3,3w h e re th e f i r s t n t e rm s a re z e ro a n d th e l a s t n t e rm s a re (C + : F ) t h u s s a t i s fy in g a s su m p t io n s

n u m b e r 1 a n d 2 d e sc r ib e d in s e c t io n 7 .1 .

E q u a t io n (1 4 ) i s a n o v e r d e t e r m i n e d l i n e a r s y s t e m ( i . e . t h e re a re m o re e q u a t io n s th a n

u n k n o w n s ) i n w h i c h t h e t e n s o r A i s d e t e rm in e d f ro m th e m e a s u re d f a u l t n o rm a l a n d s l i p v e c to r

0o r i e n t a t i o n s a n d t h e v e c t o r F r e p r e s e n t s t h e c h o s e n v a l u e s f o r : a n d C . T h e s t r e s s v e c t o r D i s

d e t e rm in e d b y u s in g a s t a n d a r d l e a s t - s q u a re s l i n e a r i n v e r s io n m e th o d (S c h ie d , 1 9 6 8 ; A n to n ,

1 9 8 1 , p . 3 1 5 -3 2 7 ; F rö b e rg , 1 9 8 5 , p . 1 5 5 -1 5 7 ,2 5 0 -2 5 4 ) . S in c e D y i e ld s t h e p a l e o s t r e s s t e n s o r

i,j 1 2 3F , t h e m a g n i tu d e s a n d o r ie n t a t io n s o f th e p r in c ip a l s t r e s s e s F , F , a n d F m a y th u s b e

d e te rm in e d .

T h e p ro g ra m u t i l i z i n g R e c h e s ' m e th o d o f p a l e o s t r e s s a n a ly s i s r e q u i r e s s ix i t e m s o f

in f o rm a t io n t o ru n - - t h e fa u l t ' s s t r i k e (0 ° 6 3 6 0 ° ) , t h e f a u l t ' s d i p a n g le ( 0 ° 6 9 0 ° ) , t h e t r e n d o f

a s l i c k e n l in e o n t h e f a u l t s u r f a c e ( 0 ° 6 3 6 0 ° ) , t h e p l u n g e o f t h a t s l i c k e n l i n e ( 0 ° 6 9 0 ° ) , t h e

ro t a t i o n s e n s e o f th e f a u l t a s v i e w e d d o w n th e p lu n g e o f th e ro t a t i o n a x i s (1 = c lo c k w is e a n d

2 = c o u n t e r c l o c k w i s e ) , a n d t h e c o n f i d e n c e l e v e l o f t h e f a u l t ( 1 = e x c e l l e n t 6 4 = p o o r ) .

T h i s i n fo rm a t io n m u s t b e in a n A S C II d a t a f i l e w h e re e a c h fa u l t d a tu m i s w r i t t e n o n a

s in g le l i n e a s in te g e rs s e p a ra te d b y b la n k s p a c e s .

F ig u r e 7 - 1 - L o w e r -h e m i s p h e re s t e r e o g ra p h i c p ro j e c t io n s h o w in g a c o n ju g a t e s e t o f tw o

n o rm a l f a u l t s w i th a n e a s t -w e s t s t r i k e a n d a d ip o f 4 5 °

A s a n e x a m p l e , a c o n j u g a t e s e t o f t w o n o r m a l fa u l t s w i t h a n e a s t - w e s t s t r ik e a n d a d i p

o f 4 5 ° ( f i g u re 7 -1 ) w o u ld b e w r i t t e n to t h e i n p u t f i l e a s

0 9 0 4 5 1 8 0 4 5 2 1

2 7 0 4 5 0 0 0 4 5 2 1

s in c e t h e f i r s t f a u l t h a s a s t r i k e o f 0 9 0 ° , a d ip o f 4 5 ° , a s l i c k e n l in e w i th a t r e n d o f 1 8 0 ° a n d a

p l u n g e o f 4 5 ° , a c o u n t e r c l o c k w i s e ( 2 ) r o t a t i o n s e n s e , a n d a c o n f i d e n c e l e v e l o f 1 ( e x c e l l e n t ) .

T h e s e c o n d f a u l t h a s a s t r i k e o f 2 7 0 ° , a d ip o f 4 5 ° , a s l i c k e n l in e w i th a t r e n d o f 0 0 0 ° a n d a

p l u n g e o f 4 5 ° , a c o u n t e r c l o c k w i s e ( 2 ) r o t a t i o n s e n s e , a n d a c o n f i d e n c e l e v e l o f 1 ( e x c e l l e n t ) .

O n c e t h e A S C I I d a t a f i l e h a s b e e n r e a d i n t o t h e p r o g r a m , t h e u s e r i s a s k e d t o s u p p l y

0 H20v a lu e s fo r th e c o e f f i c i e n t o f f r i c t i o n : , t h e c o h e s i o n C , a n d t h e f lu i d p r e s s u r e P o n t h e f a u l t .

T h e f l u id p re s s u re t e rm s im p ly re s u l t s i n a l o w e r e f f e c t iv e n o rm a l s t r e s s o n th e f a u l t a n d w a s

s e t to z e r o f o r a l l t e s t s o f t h e p r o g r a m . T h e c o e f f i c i e n t o f f r i c t i o n a n d c o h e s i o n w e r e n o r m a l l y

s e t e q u a l to th e v a lu e s u se d w h e n c r e a t in g th e a r t i f i c i a l f a u l t p o p u la t i o n s w i th th e s l i p v e c to r

c a lc u la t i o n p ro g ra m (c h a p te r 5 ) .

W h e n a l l o f th e d a t a h a s b e e n e n te r e d in to th e p ro g ra m , th e d i r e c t i o n c o s i n e s o f th e

n o r m a l a n d s l i p v e c t o r s ( e q u a t i o n s 1 a n d 2 ) a r e c a l c u l a t e d . U s i n g e q u a t i o n ( 6 ) , t h e c o m p o n e n t s

o f v e c t o r B o r th o g o n a l t o v e c t o r s N a n d S a r e d e t e rm in e d . N e x t , t h e c o e f f i c i e n t s o f m a t r ix A

i i i 0( e q u a t i o n 1 3 ) a n d v e c t o r F a re c a lc u la te d u s in g N , S , B , : , a n d C .

T h e o v e rd e t e rm in e d s y s t e m A x D = F ( e q u a t i o n 1 4 ) m u s t n o w b e s o l v e d f o r D b y u s in g

s t a n d a r d l i n e a r a l g e b r a m e t h o d s f o r d e t e r m i n i n g a l e a s t - s q u a r e s l i n e a r i n v e r s i o n o f A (S c h ie d ,

1 9 6 8 ; A n t o n , 1 9 8 1 , p . 3 1 5 - 3 2 7 ; F r ö b e r g , 1 9 8 5 , p . 1 5 5 - 1 5 7 ,2 5 0 - 2 5 4 ) s u c h t h a t

D = A x F (1 7 )-1

3,3W h e n t h e f iv e c o m p o n e n t s o f v e c t o r D a r e c a l c u la t e d , t h e v e r t i c a l s t r e s s F i s s e t e q u a l

1,1 2,2t o 1 .0 a n d s c a l e s t h e m a g n i t u d e s o f F a n d F s i n c e t h e f i r s t tw o c o m p o n e n t s o f v e c t o r D a r e

1,1 3,3 2,2 3,3(F - F ) a n d (F - F ) .

i,jT h e c o m p o n e n t s o f v e c t o r D c o n ta in a l l o f t h e F c o m p o n e n t s o f t h e s t r e s s t e n s o r F

w h ic h m a y , in tu rn , b e u se d to d e te rm in e th e m a g n i tu d e s a n d o r i e n ta t i o n s o f t h e t h re e p r in c ip a l

1 2 3s t r e s s a x e s F , F , a n d F . T h e t h r e e e i g e n v a l u e s a n d e i g e n v e c t o r s o f th e 3 x 3 s t r e s s t e n s o r F

c o r re s p o n d to th e m a g n i tu d e s a n d o r i e n t a t i o n s o f t h e t h re e p r in c ip a l s t r e s s a x e s re s p e c t iv e ly .

T h e se e ig e n v a lu e s a n d e ig e n v e c to rs m a y be c a lc u la te d u s in g s ta n d a rd l i n e a r a lg e b ra t e c h n iq u e s

(A n to n , 1 9 8 1 , p . 2 6 1 -2 8 4 ; F rö b e rg , 1 9 8 5 , p . 2 2 -2 6 ) .

i,jT h e s t r e s s t e n s o r c o m p o n e n t s F a r e n o w s u b s t i t u t e d i n t o e q u a t i o n ( 1 1 ) f o r e a c h f a u l t .

nT h e p r o g r a m c a l c u l a t e s , fo r e a c h o f th e n f a u l t s , t h e n o r m a l s t r e s s F , s h e a r s t re s s i n t h e s l i p

sd i re c t io n F , c o e f f i c i e n t o f f r i c t i o n : , a n d th e m i s f i t a n g le . T h e m i s f i t a n g le , o r a n g u la r

d i v e r g e n c e , i s th e a n g l e b e t w e e n t h e o b s e r v e d s l i p d i r e c t i o n ( g i v e n b y s l i c k e n l i n e s o n t h e f a u l t

p la n e a n d th e f a u l t ' s s h e a r s e n s e ) a n d th e e x p e c te d s l i p d i r e c t i o n d e t e rm in e d b y th e c a lc u la t e d

p a le o s t r e s s t e n so r . T h e m e a n a n g u la r d iv e rg e n c e a n d c o e f f i c i e n t o f f r i c t i o n a re a l s o c a lc u la t e d

fo r t h e p o p u la t i o n .

T h e g o a l o f R e c h e s ' m e th o d i s t o f i n d a g e o lo g ic a l ly - r e a s o n a b le c o e f f i c i e n t o f f r i c t i o n

: w h ic h w i l l r e s u l t i n a p a l e o s t r e s s t e n so r w h ic h y ie ld s t h e l o w e s t a v e ra g e a n g u la r d iv e rg e n c e .

T h e p ro g ra m re s u l t s , a lo n g w i th th e i n i t i a l d a t a , a re d i s p la y e d o n th e c o m p u te r 's s c re e n

F ig u r e 7 - 2 - F i g u r e s h o w i n g t h e g r a p h i c a l o u t p u t f ro m R e c h e s ' m e t h o d o f p a l e o s t r e s s a n a l y s i s

u s in g K e n n e t h H a rd c a s t l e ' s (1 9 8 9 ) p ro g ra m . T h i s i s a n a n a l y s i s f ro m a p o p u la t i o n o f 1 2

a r t i f i c ia l l y g e n e ra te d fa u l t s . S e e t e x t f o r a n e x p la n a t i o n o f t h e h e a d e r a b b re v ia t i o n s u s e d .

in th e fo rm a t s h o w n in f i g u re 7 -2 .

T h e o u tp u t d a t a c o n s i s t s o f - - t h e f i l e n a m e c o n t a in in g th e o r ig in a l f a u l t p o p u la t i o n

H2O 0i n p u t d a t a ; t h e f l u id p r e s s u re P , c o e f f i c i e n t o f f r i c t i o n : , a n d c o h e s io n C v a lu e s c h o se n b y

th e u s e r ; t h e n u m b e r (N ) o f f a u l t s i n th e p o p u la t i o n ; th e t r e n d a n d p lu n g e in d e g r e e s o f th e

1 2 3t h r e e p r in c ip a l s t r e s s a x e s F , F , a n d F a n d th e i r r e l a t i v e m a g n i tu d e s ; th e s t r e s s r a t i o M

(d e n o t e d b y S 1 - S 3 /S 2 - S 3 ) ; th e e n t e r e d s t r i k e a n d d ip in d e g re e s (S t r ,D ip ) o f e a c h f a u l t ; th e

e n te re d t r e n d a n d p lu n g e in d e g re e s (A z ,P l g ) o f th e s l i c k e n l in e s fo r e a c h f a u l t ; th e e n te re d

c o n f id e n c e l e v e l (C o n fd n c ) fo r e a c h fa u l t ; t h e f a u l t c l a s s o b ta in e d f ro m th e e n t e r e d ro t a t i o n

s e n s e f o r e a c h f a u l t ; t h e p r e d i c t e d f a u l t c l a s s f o r e a c h f a u l t ; th e p r e d i c t e d t re n d a n d p l u n g e i n

d e g r e e s ( A z ,P l g ) o f th e s l i c k e n l i n e s f o r e a c h f a u l t ; t h e a n g u l a r d i v e r g e n c e i n d e g r e e s ( A n g l r

D v rg n c ) b e tw e e n th e e n te re d s l ic ke n l in e o r ie n ta t io n s a n d th e p re d i c t e d s l i c k e n l in e o r i e n t a t i o n s

fo r e a c h f a u l t ; th e c a lc u la t e d n o rm a l a n d sh e a r s t r e s s e s o n e a c h fa u l t ; th e c a lc u la t e d

c o e f f i c i e n t o f f r i c t i o n ( C o e ff . F rc t 'n ) f o r e a c h f a u l t ; a n d th e a v e ra g e s o f th e c a lc u la t e d a n g u la r

d i v e r g e n c e s , n o r m a l s t re s s e s , s h e a r s t re s s e s , a n d c o e f f i c i e n t s o f f r i c t i o n f o r t h e f a u l t

p o p u la t i o n .

A f t e r t h e d a t a h a s b e e n d i s p l a y e d , t h e u s e r h a s t h e o p t i o n o f s a v i n g t h e d a t a t o a n A S C I I

d a ta f i l e a n d /o r ru n n in g th e s a m e p o p u la t i o n a g a in fo r a d i f f e re n t s e t o f f l u id p re s s u re ,

c o e f f i c i e n t o f f r i c t i o n , a n d c o h e s io n v a lu e s . T h e d a ta fo r e a c h p a le o s t r e s s a n a ly s i s I o b t a in e d

w a s p lo t t e d o n lo w e r -h e m is p h e re s te r e o g ra p h i c p ro j e c t io n s b y a T u rb o P a s c a l v e r s io n 3 .0 1

p ro g ra m I w ro te fo r th a t p u rp o s e (A p p e n d ix D ) . T h e p ro g ra m r e a d t h e f a u l t p o p u la t i o n d a t a a n d

th e p re d i c t e d p r in c ip a l s t r e s s a x e s o r i e n t a t i o n s a n d c r e a t e d A u to C A D s c r ip t f i l e s fo r p lo t t i n g

s te re o n e ts v ia A u to C A D .

R e c h e s ' m e t h o d o f p a l e o s t r e s s a n a l y s i s w a s c h o s e n fo r t e s t in g fo r s e v e r a l re a s o n s . I h ad

a u s e r - f r i e n d ly c o p y o f th e p ro g ra m w h ic h w o rk e d c o r re c t ly , t h e p ro g r a m w a s w r i t t e n fo r a n

I B M P C o r c o m p a t i b l e c o m p u t e r , t h e u s e o f a n o n - l in e a r in v e r s i o n a l l o w s t h e p r o g r a m t o

p e r fo rm c a l c u la t i o n s f o r r e a so n a b ly -s i z e d f a u l t p o p u la t i o n s i n s h o r t a m o u n ts o f t im e ( l e s s th a t

1 m in u te f o r m o s t p o p u la t i o n s ) , t h e p ro g ra m i s c l a im e d t o b e a n i m p ro v e m e n t o v e r A n g e l i e r ' s

c o m p u t a t i o n a l m e t h o d o f p a l e o s t r e s s a n a l y s i s s i n c e i t t a k e s i n t o a c c o u n t th e C o u l o m b f a i l u r e

c r i t e r i o n , a n d th e m e th o d h a s b e e n u s e d in s e v e ra l p u b l i s h e d f i e ld s tu d i e s (R e c h e s , 1 9 8 7 ;

H a rd c a s t l e , 1 9 8 9 ) .

C H A P T E R 8

P A L E O S T R E S S A N A L Y S IS T E S T D A T A

T h e a r t i f ic ia l fa u l t p o p u la t io ns u se d fo r te s t in g th e tw o p a le o s t r e s s a n a ly s i s p ro gra m s

w e re c re a ted w i th fo u r sp e c i f ic q u e s t io n s in m ind :

1 . H o w a c c u ra te a re p a leo s t re ss a n a lys i s p ro g ra m s g ive n a g e o lo g ica l ly -re a l i s t ic

p o p u la t io n o f fa u l t s?

2 . H o w d o the p a l eo s t r e s s a n a lys i s p ro gra m s r ea c t to sp ec ia l typ es o f fau l t p o p u la t io ns?

3 . H o w se ns i t iv e a re p a le o s t r e ss a n a lys is p ro g ra m s to e r ro r s in th e in i t ia l fa u l t p o p u la t io n

d a ta ?

4 . H o w d o th e r e su l t s o f p a le o s tr e ss a n a lys is p r o g r a m s c o m p a r e to o n e a n o th e r fo r th e

sa m e in i t ia l p o p u la t io n o f fa u l t s?

T o a n s w e r th e s e q u e s t io n s , th e fa u l t p o p u la t io n s d is c u ss e d in th e fo l lo w in g se c t io n s

w e re c re a ted .

8 .1 C r ea t in g th e A r t i f ic ia l F a u l t P o p u la t io n s

F o r c re a t in g t h e a r t i f ic ia l fa u l t p o p u la t io n s , a s ta n d a rd s t re ss f i e ld w a s d e f ine d . T h is

a l lo w ed fo r e a sy c o m p a r iso n b e tw ee n th e r e su l t s o f th e p a le o s t r e ss a n a lyse s a n d d id no t

s ig n i f i c a n t ly c o n s tr a in th e te s ts in a n y w a y . T h e s l ip v e c to r c a lc u la t io n p r o g r a m r e q u i r e s th e

1 3use r to sp ec i fy the o r i en ta t io ns fo r the p r inc ip a l s t r e s s ax es F a n d F , th e r e la t iv e m a g n itu d e s

1 3fo r th e m o s t c o m p re ss iv e F a nd th e le a s t c o m p re ss iv e F p r in c ip a l s t r e ss a x e s , a n d va lu e s fo r

0th e c o e f f ic ie n t o f f r ic t io n : a n d c o h e s io n C a c t ing up o n the fau l t p lan e s (c h a p te r 5 ) . F o r a l l

o f th e a r t i f ic ia l fa u l t p o p u la t io n s d i sc u sse d in th is c h a p te r , t h e a b o v e v a lu e s w e r e s e t su c h th a t

1 3F ha d a p lun g e a n d t re n d o f 9 0 /0 0 0 ( u p ) w i th a re la t ive m a g n i tud e o f + 1 .0 , F h a d a p l u n g e a n d

t r e n d o f 0 0 / 0 0 0 ( n o r t h ) w i th a r e la t iv e m a g n i tu d e o f -1 .0 , th e c o e f f ic ie n t o f f r ic t io n : w a s se t

0to 0 .8 5 ( B a r to n a n d C h o u b e y , 1 9 7 7 ; B ye r le e , 1 9 7 8 ) , a n d th e c o h e s io n C w a s se t to 0 .0 . N o te

1 3 2tha t th e o r ie n ta t io n s o f F a n d F im p ly th a t F h a s a n o r ie n ta t io n o f 0 0 /0 9 0 ( e a s t ) . T h e

2 1 3 2 3m a g n i tud e o f F i s d e p e nd a n t up o n th e s t r e ss r a t io M w h ic h i s [ (F - F ) / (F - F ) ] .

F o r e ac h fa u l t - s l ip d a tu m g en e ra te d b y th e s l ip v e c to r ca lc u la t io n p r o g ra m , th e sh e a r

s t r e ss to no rm a l s t r e s s r a t io w a s e x a m in e d . I f th i s r a t io w a s to o lo w , th e fa u l t c o u ld no t b e

e x p e c ted to s l ip u n d e r an y re a l i s t ic g e o lo g ica l c o n d i t io n s . O n ly tho se fau l t s w i th a su f f ic ien t ly

h igh sh e a r s t re ss to no rm a l s t re ss r a t io w e re us e d in a l l o f th e a n a lyse s .

T h e fa u l t o r ie n ta t io n d a ta ( th e p l u n g e a n d t r e n d o f th e fa u l t 's n o r m a l v e c to r a n d th e

p i tc h a n g le o f th e s l ip ve c to r s ) fo r th e fa u l t p la n es in e a ch p o p u la t io n d i sc u sse d in th is c ha p te r

a re l is t e d in A p p e n d ix B .

8 .2 R a n d o m F a u lt -S l ip P o p u la t io n s

T o tes t th e a c c u ra c y o f th e p a leo s t re ss a n a lys i s p ro g ra m s , th r e e ra n d o m p o le fau l t -s l ip

p o p u la t io n s w e r e c r e a te d in th e fo l lo w in g m a n n e r :

1 . A s ta n d a rd s t re ss f i e ld w a s d e c id e d up o n ( se c t io n 8 .1 ) .

2 . A T u rb o P a sc a l ve rs io n 3 .0 1 p ro g ra m w a s w r i t t e n to ra n d o m ly re tu r n two nu m b e rs - -

o n e b e tw ee n 0 a nd 8 9 in c lu s iv e a nd o n e b e tw ee n 0 a nd 3 5 9 in c lu s iv e . T h e r a nd o m

n u m b e rs w e re c h o se n th r o u g h T u rb o P a sc a l 's R A N D O M fun c t io n .

3 . T h e two nu m b e rs ge n e ra ted w e re t ak e n to b e the p lun g e a n d t re n d re sp e c t ive ly o f a p o le

(n o rm a l ve c to r ) to a fau l t p lan e .

4 . T h e p o le t o th is fa u l t p la n e w a s e n te re d in to th e s l ip v e c to r ca lc u la t io n p r o g ra m

( c h a p te r 5 ) a n d th e c a lc u la te d p i tc h a n g le s o f th e s l ip v e c to r in th e fa u l t p l a n e a n d th e

s h e a r s t r e ss to n o r m a l s t r e ss r a t io s a c t in g u p o n th e fa u l t p l a n e w e r e r e tu r n e d fo r f iv e

v a lue s o f M (M = 0 .0 0 , M = 0 .2 5 , M = 0 .5 0 , M= 0 .7 5 , a n d M = 1 .0 0 ) .

5 . I f the sh e a r s t re ss to n o rm a l s t re ss ra t io s fo r ea c h va lue o f M a c t in g u p o n th e fa u l t p l a n e

w e re a l l a b o v e a c u to f f va lue o f 1 .0 , th e f a u l t c o u ld b e e x p e c ted to s l ip un d e r re a l i s t ic

g e o lo g ica l co n d i t io n s a n d the fau l t p lan e w a s inc lud e d i n to th e ra n d o m p o le fau l t

p o p u la t io n . I f o ne o r m o re o f th e r a t io s w e r e b e lo w th e c u to ff va lu e , th e fa u l t w a s no t

inc lud e d in the p o p u la t io n .

6 . S t e p s 2 th r o u g h 5 w e r e r e p e a te d u n t i l t h r e e p o p u la t io n s o f 1 8 fa u l t p l a n e s e a c h w e r e

c re a ted .

W h i le th e fau l t p lan e s in th e ra n d o m -p o le fau l t p o p u la t io n s w e re ra n d o m ly ch o se n , i t

i s im p o r ta n t t o r e m e m b e r th a t th e y d o n o t n e c e s s a r i ly h a v e a r a n d o m sp a t ia l d i s t r ib u t io n . In

a d d t io n , o n l y th o s e ra n d o m ly-c h o s e n fa u l t s w h ic h s a t i s f ie d a fa i lu r e c r i te r ia fo r th e s ta n d a r d

s tr e ss f ie ld ( se c t io n 8 . 1 ) w e r e i n c lu d e d i n e a c h p o p u la t io n . T h e p u r p o s e o f r a n d o m ly c h o o s in g

a p lun ge an d t r en d fo r ea ch fau l t p lan e wa s no t to in su r e a r an d o m sp a t ia l d is t r ibu t io n o f fau l t s ,

b u t to insu re tha t an y b ia s in se lec t ing fau l t s to inc lud e in e a c h p o p u la t io n w a s e l im ina ted .

T h r e e p o p u la t io n s o f r a n d o m -p o le fa u l t s ( la b e l le d R P -0 1 , R P -0 2 , a n d R P -0 3 ) , f o r f iv e

v a lue s o f M e a ch , y ie ld s a to ta l o f 1 5 fa u l t p o p u la t i o n s ( f ig u re s 8 -1 6 8 -1 5 ) to te s t fo r e a ch o f

th e tw o p a le o s tr e ss a n a lys is p r o g r a m u se d . B y c o m p a r i n g t h e r e s u l t s o f th e s e te s ts to th e

o r ig in a l s ta n d a rd s t r e ss f ie ld ( se c t io n 8 .1 ) , th e a c cu ra c y o f th e p a le o s t r e ss p r o g ra m s w a s

e v a lua ted .

F ig u r e 8 - 1 - L o w e r -h e m is p h e r e s te r e o g r a p h ic p r o j e c t io n o f fa u l t p o p u la t io n R P -0 1 sh o w in g

1 8 n o r m a l f a u l t s a nd th e ir a sso c ia te d s l ip v e c to r s ( c i r c le s ) fo r a M v a lu e o f 0 . 0 0 ( R P -0 1 -0 0 ) .

1 2 3T he o r i en ta t io ns o f the th re e p r inc ip a l s t r e s s ax es F , F , a n d F a re up , ea s t , an d no r th

re sp e c t ive ly .

F ig u r e 8 -1 0 - L o w e r -h e m is p h e r e s te r e o g r a p h i c p r o j e c t io n o f fa u l t p o p u la t io n R P -0 2 sh o w in g

8 .3 C r ea t in g S p e c ia l -C a se F a u l t P o p u la t io n s

T h e b e h a v io r o f p a le o s t r e ss a na lys is p r o g ra m s to sp e c ia l -c a se fa u l t p o p u la t io n s w a s

n e x t e x am in e d . H a rd c a s t le (1 9 9 0 ) c la im e d th a t R e c he s ' m e th o d (c h ap te r 7 ) r e tu rn e d

in c o n sis te n t r e su l t s w i th s im p le co n ju ga te fa u l t p o p u la t io n s a nd A l lm e n d in ge r (1 9 8 9 ) in d ic a te d

th a t G e p h a r t a nd F o r sy th 's m e th o d ( ch a p te r 3 ) g a ve in c o n sis te n t r e su l t s fo r th ru s t fa u l t s w he n

th e y w e r e a l l o f a v e r y s im i la r o r ie n ta t i o n . G iv e n th e m a th e m a t ic s in v o lv e d in c a lc u la t in g

p a leo s t r e s s ten so r s f ro m fau l t d a t a , fau l t s wh ich ha ve o r i en ta t io ns c lo se to the p r inc ip a l s t r e s s

p la n e s ( th e p la n e s n o r m a l to th e t h r e e p r in c ip a l s t r e ss a x e s ) m a y in f lu e n c e th e c a lc u la t io n s

a d v e rse ly .

W i th the se tho u g h ts in m ind , se v e ra l sp e c ia l -ca se fau l t p o p u la t io n s w e re c re a ted - -

s im p le A nd er so n ian co n ju ga te f au l t s , o r tho rho m b ic sym m et ry fa u l t s , r ad ia l sym m et ry fau l t s ,

a n d fa u l t s w h ic h a l l h a v e a p p r o x i m a t e ly th e s a m e o r ie n ta t io n .

8 .4 A n d er so n ia n C o n ju g a te F a u l t P o p u la t io n s

C o nju ga te f au l t s e t s a r e co m m o nly fo un d in the f ie ld an d used to e s t im a te p a l eo s t r e s s

o r ie n ta t io n s (D a v is , 1 9 8 4 , p . 3 0 6 ; R a ga n , 1 9 8 5 , p . 1 3 5 ; S up p e , 1 9 8 5 , p . 2 9 2 ; R o w la nd , 1 9 8 6 ,

p . 1 3 4 ; D e n n is , 1 9 8 7 , p . 2 3 6 ; M a r sh a k a n d M it r a , 1 9 8 8 , p . 2 6 1 ; S p e n c e r , 1 9 8 8 , p . 1 9 9 ) . U s in g

A n d e rso n 's (1 9 5 1 ) th e o ry o f fa u l t in g ( se c t io n 2 .1 ) , i t i s a t r iv ia l m a t te r to a ss ig n p o ss ib le

p r in c ip a l s t r e ss a x is o r i e n t a t io n s fo r a c o n j u g a te fa u l t p o p u la t io n . T h e m o s t c o m p r e ss iv e

1p r inc ip a l s t r e s s ax is F b i se c t s the a c u te a n g le o f th e c o n ju g a te fau l t s , the in te rm e d ia te

2p r inc ip a l s t r e s s ax is F i s p a ra l le l to the in t e r se c t io n o f t h e co n ju ga te f au l t s , an d the lea s t

3co m p res s i v e p r in c ip a l s t r e s s ax is F b i se c ts th e o b tu se an g le o f th e c o n ju ga te fa u l t s . T w o

c o n ju g a te fa u l t p o p u la t io ns w e re c h o se n fo r t e s t in g to se e i f th e p a le o s t r e s s a n a lys is p ro gra m s

re tur n e d the sa m e p r in c ip a l s t re ss a x e s a s A n d e rso n 's the o ry .

T h e f i r s t c o n ju g a te fa u l t p o p u la t io n ( A C -0 1 ) , w a s c re a te d su c h th a t t h e s t r ik e s o f a l l o f

3th e fa u l t s w e re e i th e r p a ra l le l , o r su b p a r a l le l (± 5 ° ) , to th e p r in c ip a l p la n e fo r th e F a x i s w i th

1the i r a c u te a n g le b e ing b i se c ted b y F ( f ig u re s 8 -1 6 6 8 -2 0 ) . T h is p o p u la t io n i s c o n s i s te n t wi th

1 2 3A n d e r s o n 's th e o r y g iv e n th e s ta n d a r d s t r e ss f ie ld o f u p , e a s t , a n d n o r th fo r F , F , a n d F

T h e se c o n d c o n ju g a te fau l t p o p u la t io n ( A C -0 2 ) , w a s c re a ted su c h tha t a l l o f th e fau l t s

2 3w e re o r ie n te d a t 4 5 ° f ro m th e p r in c ip a l p la n es fo r th e F a n d F a x e s w i th the i r a c u te a n g le

1b e ing b i se c ted b y F ( f ig u re s 8 -2 1 6 8 -2 5 ) . T h i s po p u la t io n i s n o t c o n s is te n t w i th A n d e r so n 's

2 3th e o ry s in c e F a n d F a re o r ie n ted a t 4 5 ° f ro m the i r p re d ic ted p o s i t io n s .

F ig u r e 8 -1 6 - L o w e r -h e m is p h e r e s te r e o g r a p h i c p r o j e c t io n o f fa u l t p o p u la t io n A C -0 1 sh o w in g

1 8 n o r m a l f a u l t s a n d th e ir a sso c ia te d s l ip v e c to r s ( c i r c le s ) fo r a M v a lu e o f 0 . 0 0 ( A C -0 1 -0 0 ) .

8 .5 O r th o r h o m b ic F a u l t P o p u la t io n s

O rth o rh o m b ic , o r rh o m b o h e d r a l , fa u l t p a t te rn s h av e b e e n d e sc r ib e d in se v e ra l a re a s

(A yd in a n d R e c h e s , 1 9 8 2 ; K ra n tz , 1 9 8 6 ; K ra n tz , 1 9 8 9 ) . T h e se fau l t s a r e t e r m e d o r th o rh o m b ic

s in c e th e y a r e a r ra n g e d in a n o r th o r h o m b ic s ym m e tr y a b o u t th e p r in c ip a l s t r a in a x e s a n d

u su a l ly c o n s i s t o f tw o se t s o f c o n ju g a te no rm a l fau l t s . T h e o c c u re n c e o f the se fau l t s is n o t we l l

e x p la ine d b y c o n ju g a te fau l t in g the o ry a n d K ra n tz (1 9 8 9 ) su g g e s te d tha t o r th o rh o m b ic fau l t

p a t te r n s r e p r e s e n t th e g e n e r a l , th r e e -d im e n s io n a l s t r a i n c a s e ( th e o d d -a x is m o d e l ) a n d

c o n ju g a te fau l t s a r e thu s re leg a ted to the sp e c ia l ca se o f p lan e s t ra in .

A n o r th o r h o m b i c fa u l t p o p u la t io n c o n s is t in g o f 2 0 fa u l t s w a s c r e a te d fo r te s t in g th e

p a le o s t r e ss a na lys is p r o g ra m s (O S -0 1 ) . T h e 2 0 fa u l t s o f t h e p o p u la t io n fo r m e d f iv e d i s t in c t

se ts o f o r th o rh o m b ic fa u l t s w i th ± 5 ° o f fse ts in s t r ik e a nd d ip f ro m th e tw o c o n ju g a te se ts a t 4 5 °

2 3 1f ro m th e F a n d F p r in c ip a l p lan e s . F o r a l l o f th e c o n ju g a te f a u l t se t s in the p o p u la t io n , F

2 3b ise c te d th e ir a c u te a ng le a nd F o r F b ise c te d th e ir o b tu se a ng le s ( f ig u re s 8 -2 6 6 8 -3 0 ) .

F ig u r e 8 -2 6 - L o w e r -h e m is p h e r e s te r e o g r a p h i c p r o j e c t io n o f fa u l t p o p u la t io n O S - 0 1 sh o w in g

2 0 o r th o rh o m b ic no r m a l fa u l t s a n d th e ir a sso c ia te d s l ip v e c to r s ( c i r c le s ) fo r a M v a lu e o f 0 .0 0

1 2 3(O S -0 1 -0 0 ) . T he o r i en ta t io ns o f the th re e p r inc ip a l s t r e s s ax es F , F , a n d F a r e u p , e a s t , a n d

n o r th re sp e c t ive ly .

8 .6 R a d ia l S y m m etr y F a u l t P o p u la t io n s

W h ile c re a t in g th e c o n ju ga te an d o r t ho r ho m b ic fa u l t p o p u la t io n s , I n o t ic e d th a t

c o n ju g a te no rm a l fa u l t se t s o r ie n ted a t a w id e ra n g e o f s t r ik e d i re c t io n s w o u ld e x p e r ie n c e s l ip

in m y s ta n d a rd s t re ss f i e ld ( se c t io n 8 .1 ) . T h e o n ly ex c e p t io n s w e re tho se c o n ju g a te se t s w i th

1a s t r ik e o f le ss t h a n 3 0 ° f r o m th e F p r in c ip a l s t r e ss d i re c t io n . I th e re fo r e c re a te d a ra d ia l

sym m et ry fau l t p o p u la t io n (R S -0 1 ) co ns i s t ing o f 1 0 no rm a l fau l t s w i th s t r ikes o f 3 0 ° , 6 0 ° , 9 0 ° ,

11 2 0 ° , a n d 1 5 0 ° fo r e a c h o f th e f iv e c o n ju g a te se t s . T h e i r a c u te a n g le s w e re b i se c ted b y F

( f ig u re s 8 -3 1 6 8 -3 5 ) . W h ile th is m a y n o t b e a g e o lo g ic a l ly - r e a so n a b le fa u l t p o p u la t io n , i t d o e s

sa t i s fy a l l o f th e in i t i a l p a leo s t re ss a ssu m p t io n s a n d sh o u ld b e s o lva b le b y the a n a lys i s

p ro g ra m s . F o r th i s re a so n , th e p o p u la t io n w a s inc lud e d in the s tud y .

8 .7 F a u l t P o p u la t io n s o f a S im i la r O r ie n ta t io n

F ina l ly , th r e e sp e c ia l -ca se fau l t p o p u la t io n s w e re c re a ted s u c h t h a t a l l o f th e fau l t s in

e ac h p o p u la t io n ha d a v e ry s im i la r o r ie n ta t io n (S O -0 1 , S O -0 2 , a n d S O -0 3 ) . T h is i s a typ e o f

fau l t p o p u la t io n w h ich m a y re a so n a b ly b e e x p e c ted to fo rm .

T h e f i r s t p o p u la t io n ( S O -0 1 ) o f 1 5 n o r m a l fa u l t s w a s c r e a te d su c h th a t th e s t r ik e s w e r e

3p a r a l le l , a n d s u b p a r a l l e l (± 3 ° a nd ± 6 ° ) , to th e F p r inc ip a l p lan e . A l l o f the fau l t s we re a l so

1o r ie n te d su c h th a t th e y m a d e a n a n g le o f 3 0 ° (± 5 ° ) f r o m th e F p r in c ip a l s t r e ss d i r e c t io n

( f ig u re s 8 -3 6 6 8 -4 0 ) .

T h e se c o n d p o p u l a t io n ( S O -0 2 ) o f 1 5 n o r m a l fa u l t s w a s c re a te d su c h th a t th e ir s t r ik e s

2w e re o r ie n te d a t 3 0 ° ( ± 3 ° a n d ± 6 ° ) f ro m th e F d i r e c t io n . A l l o f th e fa u l t s w e re a l so o r ie n te d

1su c h th a t th e y m ad e a n a n g le o f 3 0 ° ( ± 5 ° ) f r o m th e F p r in c ip a l s t r e ss d i r e c t io n ( f ig u re s 8 -4 1

6 8 -4 5 ) .

T h e th i rd p o p u la t io n (S O -0 3 ) o f 1 5 n o r m a l fa u l t s w a s c re a te d su c h th a t th e ir s t r ik e s

2 3w e re o r ie n te d a t 4 5 ° (± 3 ° a nd ± 6 ° ) f ro m th e F a n d F d i r ec t io ns . A l l o f t h e f a u l t s we re a l so

1o r ie n te d su c h th a t th e y m a d e a n a n g le o f 3 0 ° (± 5 ° ) f ro m th e F p r i n c ip a l s t r e ss d i r e c t io n

( f ig u re s 8 -4 6 6 8 -5 0 ) .

F ig u r e 8 -3 1 - L o w e r -h e m is p h e r e s te r e o g r a p h i c p r o j e c t io n o f fa u l t p o p u la t io n R S - 0 1 sh o w in g

1 0 ra d ia l sym m e try n o rm a l fa u l t s a nd th e ir a sso c ia te d s l ip v e c to r s ( c i r c l e s ) fo r a M v a lu e o f

1 2 30 .00 (R S -0 1 -0 0 ) . T he o r i en ta t io ns o f the th re e p r inc ip a l s t r e s s ax es F , F , a n d F a re up , ea s t ,

a n d no r th re sp e c t ive ly .

F ig u r e 8 -3 6 - L o w e r -h e m is p h e r e s te r e o g r a p h i c p r o j e c t io n o f fa u l t p o p u la t io n S O -0 1 sh o w in g

1 5 n o r m a l fa u l t s o f a p p r o x i m a t e ly th e s a m e o r ie n ta t io n a n d th e i r a ss o c i a te d s l ip v e c to r s

(c i r c le s ) fo r a M v a lu e o f 0 .0 0 (S O -0 1 -0 0 ) . T h e o r ie n ta t io n s o f th e th re e p r in c ip a l s t r e ss a xe s

1 2 3F , F , a n d F a re up , ea s t , an d no r th re sp e c t ive ly .

8 .8 F in a l F a u l t P o p u la t io n T e s ts

T o te s t th e s e n s i t iv i ty o f th e p a le o s tr e ss a n a lys is p r o g r a m s , th e fo l lo w in g s te p s w e r e

p e r fo r m e d to th e r a n d o m -p o le fa u l t p o p u la t io n s ( R P -0 1 , R P -0 2 , a n d R P -0 3 ) :

1 . R a n d o m ly r e m o v in g o n e o r m o r e p la n e s f r o m t h e p o p u la t io n a n d o b s e rv in g th e c h a n g e

in th e c a lcu la ted p a leo s t re ss t e n so r .

2 . R a n d o m ly a d d in g o n e o r m o r e p la n e s to t h e p o p u la t io n s a n d o b s e rv in g th e c h a n g e in th e

c a lcu la ted p a leo s t re ss t e n so r .

3 . G iv in g a r a n d o m ± 5 ° v a r ia b i l i ty in p l u n g e a n d t r e n d o f th e fa u l t n o r m a l v e c to r s a n d

g iv in g a ± 5 ° v a r ia b i l i ty in the p i tch an g le s o f th e s l ip ve c to rs f o r e a c h fa u l t d a tum in

the p o p u la t io n .

In th is w a y, th e se n s i t iv i ty o f th e tw o p a le o s t r e ss a na lys is p r o g ra m s to in su ff ic ie n t d a ta ,

e x t ra n e o u s d a ta , an d m e a su re m e n t e r ro rs wa s e v a lua ted .

A n ge l i e r ' s (1 9 7 9 ) f ie ld d a ta (F D -0 1 ) o f 3 8 N e o g e ne ag e n o rm a l fa u l t s f ro m C e n tra l

C re te , G re e c e ( f igu re 8 -5 1 a n d tab le 8 -1 ) w e re a l so ru n th r o u g h the p a leo s t re ss a n a lys i s

p r o g ra m s . T h is d a ta p o p u la t io n h as b e c o m e a n un o ff ic ia l s t a n d a r d a ga in s t w h ic h o th e r

p a l e o s t r e s s a na lys is m e th o d s a re te s te d to se e i f th e y y ie ld the sa m e re su l t s a s A ng e l ie r

(G e p ha r t a n d F o rsy th , 1 9 8 4 ; M ic ha e l , 1 9 8 4 ; R e ch e s , 1 9 8 7 ) a n d is o n ly u se d fo r a c o m p a r iso n

h e re - - I d id no t wo r r y a b o u t wh e the r th e re su l t s we re ge o lo g ica l ly -re a so n a b le o r n o t .

F ig u r e 8 -5 1 - L o w e r -h e m is p h e r e s te r e o g r a p h i c p r o j e c t io n o f fa u l t p o p u la t io n F D -0 1 sh o w in g

3 8 no rm a l fau l t s f ro m C en t ra l C re t e , G ree ce (A ng e l ie r , 1 9 7 9 ) an d the i r a s so c ia t ed s l ip ve c to r s .

FaultPlane Slickenline

Strike Dip Trend Plunge

045° 61° 115° 59°

036° 59° 145° 58°

270° 80° 286° 57°

232° 68° 292° 65°

225° 63° 290° 61°

290° 88° 293° 59°

254° 78° 278° 62°

046° 60° 155° 59°

257° 61° 355° 61°

067° 56° 153° 56°

049° 70° 187° 61°

216° 50° 312° 50°

058° 51° 165° 50°

079° 62° 195° 59°

236° 62° 313° 61°

214° 61° 275° 58°

034° 60° 151° 57°

037° 63° 138° 63°

068° 72° 099° 58°

049° 53° 139° 90°

189° 47° 295° 46°

237° 45° 296° 41°

112° 74° 260° 61°

205° 42° 296° 42°

214° 56° 309° 56°

037° 77° 202° 48°

057° 61° 195° 51°

248° 58° 006° 55°

061° 67° 173° 65°

028° 58° 168° 46°

030° 69° 106° 68°

041° 63° 144° 62°

023° 68° 105° 68°

249° 48° 346° 48°

248° 69° 332° 69°

195° 68° 310° 66°

274° 70° 320° 63°

267° 71° 000° 71°

T a b l e 8 - 1 - A l is t ing o f th e o r ie n ta t io n s fo r A n g e l ie r 's (1 9 7 9 ) p o p u la t io n o f 3 8 no rm a l fa u l t sf ro m C e n t ra l C re te , G re e c e .

C H A P T E R 9

T E S T IN G P R O C E D U R E S A N D R E S U L T S

In th i s c h a p te r , t h e r e s u l t s o f th e t e s t s o n th e a r t i f i c i a l f a u l t p o p u la t i o n s b y th e tw o

p a le o s t r e s s a n a ly s i s p ro g ra m (R e c h e s ' m e th o d a n d A n g e l i e r ' s m e th o d ) a re p re s e n te d .

9 .1 T e st in g P r o c e d u r e s

T h e t e s t i n g p ro c e d u r e s u se d w e re v e ry s t r a ig h t fo rw a rd . T h e a r t i f ic ia l f a u l t p o p u la t i o n s

s h o w n in c h a p te r 8 a n d l i s t e d i n a p p e n d ix B , w e re c re a te d in t h e m a n n e r d i s c u ss e d in c h a p te r

5 . T h e f a u l t - s l i p d a t a f o r e a c h p o p u l a t i o n a t e a c h o f t h e f iv e v a l u e s o f M e x a m in e d (0 .0 0 , 0 .2 5 ,

0 .5 0 , 0 .7 5 , a n d 1 .0 0 ) w e re m a n u a l ly c o n v e r t e d in to in p u t fo rm a t s c o m p a t ib l e w i th th e tw o

p r o g r a m s . T h i s n e w f a u l t d a t a w a s t h e n r e a d i n t o t h e p a l e o s t r e s s a n a l y s i s p r o g r a m s a n d r e s u l t s

w e re o b ta in e d . T h e a n g le s b e tw e e n th e k n o w n a n d th e c a lc u la t e d o r i e n ta t i o n s o f th e p r in c ip a l

1 2 3s t r e s s a x e s F , F , a n d F w e re d e te rm in e d u s in g th e p ro g ra m l i s te d in a p p e n d ix E .

T h e s e re s u l t s w e re c a re fu l ly e x a m in e d to s e e i f t h e y w e re c o n s i s t e n t w i th th e i n i t i a l

a s su m p t i o n s u s e d t o c r e a t e t h e a r t i f i c i a l f a u l t p o p u la t i o n s . I f t h e y w e re n o t th e s a m e , th e

re a s o n s w h y w e re a s c e r t a in e d (o r a t l e a s t s u rm is e d ) .

9 .2 R a n d o m - P o le F a u l t P o p u la t io n R e s u l t s

T h e th re e r a n d o m -p o le f a u l t p o p u l a t i o n s R P - 0 1 , R P - 0 2 , a n d R P - 0 3 w e re t h e f i r s t t o b e

e x a m in e d . T h e re s u l t s o f th i s e x a m in a t io n a r e s h o w n in f i g u re s 9 -1 th r o u g h 9 -1 5 fo r R e c h e s '

m e th o d a n d f i g u re s 9 -1 6 th ro u g h 9 -3 0 fo r A n g e l i e r ' s m e th o d . C o m p a r i s o n s o f th e r e s u l t s o f

R e c h e s ' a n d A n g e l i e r ' s m e th o d s a re g iv e n in t a b le s 9 -1 th ro u g h 9 -3 .

Known Program Error

1F 90/000 71/195 19.0°

2F 00/090 17/345 75.7°

3F 00/000 09/078 78.1°

M 0.00 0.069 0.069

F ig ur e 9 -1 - L o w e r-h e m is p h e re s te re o g ra p h ic p ro je c t io n a n d ta b le d e m o n s tra t in g th e

1 2 3d i ffe ren ce s be tw ee n the k no w n a nd ca lcu la t ed p r inc ipa l s t re s s ax es F , F , a n d F a n d th e

v a lue o f M fo r fa u l t p o p u la t io n R P - 0 1 -0 0 u s in g R e c h e s ' m e th o d o f p a le o s t re s s a n a ly s is . In th e

1 2 3s te reo graph ic p ro jec t ion , the kno w n p r inc ipa l s t re s s ax es F , F , a n d F a re o r ien ted up , e a s t ,

a n d n or th re sp e c t iv e ly an d th e c a lc u la te d p r in c ip a l s t re ss a xe s a re de n o te d b y th e f i l le d c i rc le s

la b e l l e d 1 , 2 , an d 3 .

Known Program Error

1F 90/000 83/205 7.0°

2F 00/090 02/100 10.2°

3F 00/000 07/010 12.2°

M 0.25 0.131 0.119

F ig u r e 9 - 2 - L o w e r -h e m is p h e re s te r e o g ra p h i c p ro j e c t io n a n d t a b l e d e m o n s t r a t i n g th e

1 2 3d i f f e r e n c e s b e t w e e n t h e k n o w n a n d c a l c u la t e d p r in c ip a l s t r e s s a x e s F , F , a n d F a n d th e

v a l u e o f M f o r fa u l t p o p u l a t i o n R P - 0 1 - 2 5 u s i n g R e c h e s ' m e t h o d o f p a l e o s t r e s s a n a l y s i s . In

1 2 3t h e s t e r e o g r a p h ic p ro j e c t io n , t h e k n o w n p r in c ip a l s t r e s s a x e s F , F , a n d F a r e o r i e n t e d u p ,

e a s t , a n d n o r th re s p e c t iv e ly a n d th e c a lc u la t e d p r in c ip a l s t r e s s a x e s a re d e n o te d b y th e f i l l e d

c i r c le s l a b e l l e d 1 , 2 , a n d 3 .

Known Program Error

1F 90/000 83/205 7.0°

2F 00/090 02/096 6.3°

3F 00/000 07/006 9.2°

M 0.50 0.387 0.113

Figure 9-3 - Lower-hemisphere stereographic projection and table demonstrating the differences

1 2 3between the known and calculated principal stress axes F , F , and F and the value of M for fault

population RP-01-50 using Reches' method of paleostress analysis. In the stereographic projection, the

1 2 3known principal stress axes F , F , and F are oriented up, east, and north respectively and the calculated

principal stress axes are denoted by the filled circles labelled 1, 2, and 3.

Known Program Error

1F 90/000 83/214 7.0°

2F 00/090 04/095 6.4°

3F 00/000 07/005 8.6°

M 0.75 0.683 0.112

Known Program Error

1F 90/000 82/217 8.0°

2F 00/090 04/095 6.4°

3F 00/000 07/004 8.1°

M 1.00 0.903 0.097

Known Program Error

1F 90/000 80/347 10.0°

2F 00/090 07/214 56.3°

3F 00/000 08/124 56.4°

M 0.00 0.178 0.178

Known Program Error

1F 90/000 83/330 7.0°

2F 00/090 03/084 6.7°

3F 00/000 06/175 7.8°

M 0.25 0.178 0.072

Known Program Error

1F 90/000 83/342 7.0°

2F 00/090 02/089 2.2°

3F 00/000 07/179 7.1°

M 0.50 0.351 0.149

Known Program Error

1F 90/000 80/321 10.0°

2F 00/090 06/090 6.0°

3F 00/000 07/180 7.0°

M 0.75 0.637 0.113

Known Program Error

1F 90/000 74/300 16.0°

2F 00/090 14/089 14.0°

3F 00/000 08/181 8.1°

M 1.00 0.903 0.128

Known Program Error

1F 90/000 88/049 2.0°

2F 00/090 02/256 14.1°

3F 00/000 01/156 14.0°

M 0.00 0.070 0.070

Known Program Error

1F 90/000 87/047 3.0°

2F 00/090 02/265 5.4°

3F 00/000 02/175 5.4°

M 0.25 0.252 0.002

Known Program Error

1F 90/000 88/076 2.0°

2F 00/090 02/266 4.5°

3F 00/000 00/176 4.0°

M 0.50 0.437 0.063

Known Program Error

1F 90/000 86/098 4.0°

2F 00/090 04/26 5.7°

3F 00/000 01/356 4.1°

M 0.75 0.658 0.092

Known Program Error

1F 90/000 69/090 21.0°

2F 00/090 21/266 21.4°

3F 00/000 01/356 4.1°

M 1.00 0.880 0.120

Known Program Error

1F 90/000 79/014 11.0°

2F 00/090 00/104 14.0°

3F 00/000 11/194 17.7°

M 0.00 0.236 0.236

population RP-01-00 using Angelier's method of paleostress analysis. In the stereographic projection, the

Known Program Error

1F 90/000 82/022 8.0°

2F 00/090 02/279 9.2°

3F 00/000 08/189 12.0°

M 0.25 0.390 0.140

Known Program Error

1F 90/000 83/059 7.0°

2F 00/090 05/277 8.6°

3F 00/000 04/186 7.2°

M 0.50 0.550 0.050

Known Program Error

1F 90/000 78/095 12.0°

2F 00/090 12/274 12.6°

3F 00/000 00/004 4.0°

M 0.75 0.694 0.056

Known Program Error

1F 90/000 66/102 24.0°

2F 00/090 24/271 24.0°

3F 00/000 04/003 5.0°

M 1.00 0.796 0.204

Known Program Error

1F 90/000 87/304 3.0°

2F 00/090 03/087 4.2°

3F 00/000 02/177 3.6°

M 0.00 0.077 0.077

Known Program Error

1F 90/000 86/323 4.0°

2F 00/090 03/092 3.6°

3F 00/000 03/183 4.2°

M 0.25 0.253 0.003

Known Program Error

1F 90/000 86/351 4.0°

2F 00/090 01/093 3.2°

3F 00/000 04/183 5.0°

M 0.50 0.443 0.060

Known Program Error

1F 90/000 83/349 7.0°

2F 00/090 02/092 2.8°

3F 00/000 07/182 7.3°

M 0.75 0.675 0.075

Known Program Error

1F 90/000 82/012 8.0°

2F 00/090 01/272 2.2°

3F 00/000 08/182 8.2°

M 1.00 0.846 0.154

population RP-02-10 using Anglier's method of paleostress analysis. In the stereographic projection, the

Known Program Error

1F 90/000 82/003 8.0°

2F 00/090 03/251 19.2°

3F 00/000 07/160 21.1°

M 0.00 0.149 0.149

Known Program Error

1F 90/000 85/357 5.0°

2F 00/090 00/262 8.0°

3F 00/000 05/172 9.4°

M 0.25 0.326 0.076

Known Program Error

1F 90/000 88/329 2.0°

2F 00/090 01/085 5.1°

3F 00/000 02/175 5.4°

M 0.50 0.528 0.028

Known Program Error

1F 90/000 86/256 4.0°

2F 00/090 04/087 5.0°

3F 00/000 01/357 3.2°

M 0.75 0.728 0.022

Known Program Error

1F 90/000 73/260 17.0°

2F 00/090 16/089 16.0°

3F 00/000 02/358 2.8°

M 1.00 0.901 0.099

F o r fa u l t p o p u l a t io n R P -0 1 , R e c h e s ' m e t h o d r e tu r n e d th e l a r g e s t p r in c ip a l s t r e s s

2 3o r i e n t a t i o n e r ro r s fo r a n i n i t i a l M v a l u e o f 0 .0 0 w h e r e t h e o r i e n t a t i o n s o f F a n d F w e r e

r e v e r s e d . T h e o r i e n t a t i o n e r ro r s w e r e a l l l e s s t h a n 1 5 ° f o r a n i n i t i a l M v a l u e o f 0 .2 5 a n d a l l

l e s s t h a n 1 0 ° f o r i n i t i a l M v a lu e s o f 0 .5 0 , 0 .7 5 , a n d 1 .0 0 . T h e e r ro rs in m a g n i tu d e b e tw e e n

t h e i n i t i a l a n d c a l c u l a t e d v a l u e s o f M w e re a l l l e s s th a n 0 .1 5 ( f i F ig u re s 9 -1 th ro u g h 9 -5 ) .

o r i e n t a t i o n e r ro r s fo r a n i n i t i a l M v a lu e o f 0 .0 0 w i th th e l a rg e s t e r ro r s i n th e o r i e n t a t i o n s o f

2 3F a n d F . T h e o r i e n t a t i o n e r ro r s w e r e a l l 1 0 ° o r l e s s f o r a l l o t h e r in i t i a l M v a lu e s . T h e

e r ro r s i n m a g n i t u t e b e t w e e n t h e i n i t i a l a n d c a l c u l a t e d v a l u e s o f M w e re a l l l e s s th a n 0 .1 8

( f iF ig u re s 9 -6 th ro u g h 9 -1 0 ) .

1 2F a n d F . T h e o r i e n t a t i o n e r ro r s w e r e a l l l e s s t h a n 1 5 ° f o r a n i n i t i a l M v a l u e o f 0 .0 0 a n d a l l

l e s s t h a n 1 0 ° f o r i n i t i a l M v a lu e s o f 0 .2 5 , 0 .5 0 , a n d 0 .7 5 . T h e e r ro rs in m a g n i tu d e b e tw e e n

t h e i n i t i a l a n d c a l c u l a t e d v a l u e s o f M w e re a l l 0 .1 2 o r l e s s ( f i F ig u re s 9 -1 1 th ro u g h 9 -1 5 ) .

F o r fa u l t p o p u l a t io n R P -0 1 , A n g e l i e r 's m e t h o d r e tu r n e d th e l a r g e s t p r in c ip a l s t r e s s

1 2F a n d F . T h e o r i e n t a t i o n e r ro r s w e r e a l l 2 0 ° o r l e s s f o r a l l o t h e r in i t i a l M v a lu e s . T h e

e r ro r s i n m a g n i t u d e b e t w e e n t h e i n i t i a l a n d c a l c u l a t e d v a l u e s o f M w e re a l l l e s s th a n 0 .2 4

( f iF ig u re s 9 -1 6 th ro u g h 9 -2 0 ) .

F o r f a u l t p o p u la t i o n R P - 0 2 , A n g e l i e r ' s m e th o d re tu rn e d p r in c ip a l s t r e s s o r i e n t a t i o n

e r ro r s o f l e s s t h a n 1 0 ° f o r a l l i n i t i a l M v a lu e s . T h e e r ro rs in m a g n i tu d e b e tw e e n th e i n i t i a l

a n d c a l c u l a t e d v a l u e s o f M w e r e a l l l e s s t h a n 0 . 1 6 f o r a n i n i t a l M v a lu e o f 1 .0 0 a n d le s s th a n

0 .0 8 f o r i n i t i a l M v a lu e s o f 0 .2 5 , 0 .5 0 , 0 .7 5 , a n d 1 .0 0 ( f i F ig u re s 9 -2 1 th ro u g h 9 -2 5 ) .

F o r fa u l t p o p u l a t io n R P -0 3 , A n g e l i e r 's m e t h o d r e tu r n e d th e l a r g e s t p r in c ip a l s t r e s s

o r i e n t a t i o n e r ro r s fo r in i t i a l M v a lu e s o f 0 .0 0 a n d 1 .0 0 w i th th e l a rg e s t e r ro r s i n th e

2 3 2 3o r i e n t a t i o n s o f F a n d F f o r a n i n i t i a l M v a lu e o f 0 .0 0 a n d F a n d F f o r a n i n i t i a l M v a lu e

o f 1 .0 0 . T h e o r i e n t a t i o n e r ro r s w e r e a l l 1 0 ° o r l e s s f o r a l l o t h e r M v a l u e s . T h e e r ro r s i n

m a g n i t u d e b e t w e e n t h e i n i t i a l a n d c a l c u l a t e d v a l u e s o f M w e re a l l l e s s th a n 0 .1 5 ( f iF i g u re s

9 -2 6 th ro u g h 9 -3 0 ) .

A s a g e n e ra l o b s e rv a t io n , t h e c a l c u l a t e d o r ie n t a t io n s o f s o m e o f th e p r in c ip a l s t r e s s

1 2 3a x e s F , F , a n d F s e e m t o b e d i s p l a c e d f ro m t h e i r t r u e o r i e n t a t i o n s s u c h t h a t th e y a r e

r o u g h l y s u b p a r a l l e l lo c a l c o n c e n t r a t i o n s o f s l ip v e c t o r s . F i g u r e 9 - 1 d i s p l a y s t h i s w e l l w i t h

1t h e o r i e n t a t i o n o f F .

T a b l e s 9 -1 , 9 -2 , a n d 9 -3 c o m p a re th e r e s u l t s o f R e c h e s ' a n d A n g e l i e r ' s m e th o d s fo r

e a c h o f t h e th re e ra n d o m -p o le fa u l t p o p u la t i o n s R P -0 1 , R P -0 2 , a n d R P -0 3 re s p e c t i v e ly .

T a b l e 9 - 1 s h o w s R e c h e s ' a n d A n g e l i e r ' s m e t h o d s t o p o o r l y c o r re s p o n d f o r fa u l t

p o p u l a t i o n R P - 0 1 g i v e n i n i t i a l M v a lu e s o f 0 .0 0 a n d 1 .0 0 w i th a 2 0 ° o r l e s s a n g u la r

d e v i a t i o n f o r i n t e r m e d i a t e i n i t i a l M v a lu e s . T a b le 9 -2 s h o w s R e c h e s ' a n d A n g e l i e r ' s

m e t h o d s t o s i m i l a r l y p o o r l y c o r re s p o n d g i v e n i n i t i a l M v a lu e s o f 0 .0 0 a n d 1 .0 0 w i th a 1 0 ° o r

l e s s a n g u l a r d e v i a t i o n f o r i n t e r m e d i a t e i n i t i a l M v a lu e s . T a b l e 9 -3 s h o w s R e c h e s ' a n d

A n g e l i e r ' s m e t h o d s t o p o o r l y c o r re s p o n d g i v e n a n i n i t i a l M v a lu e o f 1 .0 0 w i th a 1 0 ° o r l e s s

a n g u l a r d e v i a t i o n f o r a l l o t h e r in i t i a l M v a lu e s . T h e e r ro r s i n m a g n i tu d e b e t w e e n R e c h e s '

a n d A n g e l i e r ' s m e t h o d s f o r t h e i n i t i a l a n d c a l c u l a t e d v a l u e s o f M w e r e 2 .2 5 o r l e s s f o r fa u l t

p o p u la t i o n R P -0 1 a n d 0 .1 0 o r l e s s fo r fa u l t p o p u la t i o n s R P -0 2 a n d R P -3 .

T h e s e r e s u l t s g e n e r a l l y i n d i c a t e t h a t b o t h R e c h e s ' a n d A n g e l i e r ' s m e t h o d s r e t u r n

r e a s o n a b l e ( ± 2 0 ° ) p r i n c i p a l s t re s s a x i s o r i e n t a t i o n r e s u l t s fo r in t e r m e d i a t e i n i t i a l M v a lu e s

( 0 . 2 5 , 0 . 5 0 , a n d 0 . 7 5 ) b u t m a y n o t p e r fo r m w e l l g i v e n i n i t i a l M v a lu e s o f 0 .0 0 o r 1 .0 0 (p l a n e

2s t r a i n ) . T h i s i s m o s t l ik e l y d u e t o t h e f a c t th a t fo r a n i n i t i a l M v a lu e o r 0 .0 0 , F i s e q u a l in

3 2 1m a g n i tu d e to F a n d f o r a n i n i t i a l M v a lu e o f 1 .0 0 , F i s e q u a l in m a g n i tu d e to F a n d th e

p ro g ra m s h a v e d i f f i c u l t y in a s s ig n in g th e p ro p e r o r i e n ta t i o n s fo r th e s e s t r e s s a x e s .

Reches' Angelier's Deviation

1F 71/195 79/014 30.0°

2F 17/345 00/104 62.4°

3F 09/078 11/194 66.7°

M 0.069 0.236 0.167

1F 83/205 82/022 15.0°

2F 02/100 02/279 4.1°

3F 07/010 08/189 15.0°

M 0.165 0.390 0.225

1F 83/205 83/059 13.4°

2F 02/096 05/277 7.1°

3F 07/006 04/186 11.0°

M 0.378 0.550 0.172

1F 83/214 78/095 16.5°

2F 04/095 12/274 16.0°

3F 07/005 00/004 7.1°

M 0.638 0.694 0.056

1F 82/217 66/102 28.3°

2F 04/095 24/271 28.3°

3F 07/004 04/003 3.2°

M 0.903 0.796 0.107

1 2 3Table 9-1 - Table demonstrating the deviation for the orientations of the F , F , and F principal stress

axes and the principal stress magnitude ratio M between Reches' and Angelier's methods of paleostress

analysis on fault population RP-01 at five different values of M.

1F 80/347 87/304 8.1°

2F 07/214 03/087 53.8°

3F 08/124 02/177 53.1°

M 0.178 0.077 0.101

1F 83/330 86/323 3.1°

2F 03/084 03/092 8.0°

3F 06/175 03/183 8.5°

M 0.178 0.253 0.075

1F 83/342 86/351 3.1°

2F 02/089 01/093 4.1°

3F 07/179 04/183 5.0°

M 0.351 0.443 0.092

1F 80/321 83/349 5.0°

2F 06/090 02/092 4.5°

3F 07/180 07/182 2.0°

M 0.637 0.675 0.038

1F 74/300 82/012 15.5°

2F 14/087 01/272 15.8°

3F 08/081 08/182 80.3°

M 0.872 0.846 0.026

analysis on fault population RP-02 at five known values of M.

1F 88/049 82/003 6.8°

2F 02/256 03/251 5.1°

3F 01/166 07/160 8.5°

M 0.070 0.149 0.079

1F 87/047 85/357 5.6°

2F 02/265 00/262 3.6°

3F 02/175 05/172 4.2°

M 0.252 0.326 0.074

1F 88/076 88/329 3.2°

2F 02/266 01/085 3.2°

3F 00/176 02/175 2.2°

M 0.437 0.528 0.091

1F 86/098 86/256 7.9°

2F 04/266 04/087 8.1°

3F 01/356 01/357 1.0°

M 0.658 0.728 0.070

1F 69/090 73/260 37.9°

2F 21/266 16/089 37.1°

3F 01/356 02/358 2.2°

M 0.880 0.901 0.021

analysis on fault population RP-03 at five different values of M.

9 .3 Sp ec ia l-C a se F a ul t P o pu la t io n R esult s

T h e f i r s t spe c ia l -ca se fau l t p o p u la t io n s to b e t es ted w e re th e tw o c o n jug a te fau l t

p op u la t io ns A C -0 1 a n d A C -0 2 . In c o n ju ga te fa u l t p op u la t io n A C -0 1 , th e c o n ju ga te fa u l t se t w a s

3 1 2pa ra l le l to t he p r inc ipa l s t re s s ax is F an d pe rpe nd icu la r the the p r inc ipa l s t re s s ax es F a n d F .

2 3In c o n ju g a te fa u l t p o p u la t io n A C -0 2 , th e c o n ju g a te fa u l t se t w a s a t 4 5 ° to b o th F a n d F a n d

1pe rpe nd icu la r the the p r inc ipa l s t re s s ax is F .

U t i l i z ing R e c h e s ' m e tho d , c o n jug a te po p u la t ion A C -01 re tu rne d su rpr i s ing resu l t s

( f ig u re s 9 -3 1 th ro ug h 9 -3 5) . T h e M v a lue s re tu rne d w e re th e op p o s i te o f w h a t on e w o u ld e x p e c t .

A M o f 1 .0 0 w a s re tu rne d for an in i t i a l M o f 0 .0 0 , a M o f 0 .7 0 w a s r e tu rn e d for an in i t i a l M o f

0 .2 5 , a M o f 0 .4 9 w a s r e tu rn e d f o r an in i t i a l M o f 0 .5 0 (w h ic h i s q u i te go od ) , a M o f 0 .2 7 w a s

re tu rne d for a n in i t i a l M o f 0 .7 5 , a nd a M o f 0 .0 9 w as re tu rne d for an in i t i a l M o f 1 .0 0 . T h is i s

1 3m o st l ik e ly re la te d to th e fa c t th a t t h e F a n d F p rin c ip a l s t re ss a xe s w e re sw i tc h ed an d h ad

1 2o p p o s i te o r ie n ta t io n s f ro m w h a t w a s e x p e c te d ( i .e . F h a d the p rop e r o r ien ta t ion fo r F a nd v ic e

v e rsa ) . T h is is ob v io u s ly a sy s te m a ti c e rro r a r i s in g w ith in th e p ro g ra m a lg o ri th m s .

A n ge l ie r 's m e th od p ro du c ed an ex a c t m a tc h b e tw e e n th e in i t i a l a nd ca lc u la te d

o r ie n ta t io n s o f t h e th re e p r in c ip a l s t re s s a x e s fo r c o n ju g a te p o p u la t io n A C -0 1 ( f ig u re s 9 -3 6

thro u g h 9 -41 ) . T h e e r ro rs in m a g n i tud e b e tw e en th e in i t i a l a n d c a lcu la ted va lue s o f M ho w e v e r ,

w e re q u i te l a rge (ap p roa c h ing 0 .5 ) fo r in i t i a l M v a lu e s o f 0 .0 0 a n d 1 .0 0 a n d sm a l le s t (0 .0 0 4 ) fo r

a n in i t i a l M v a lu e o f 0 .50 .

C o n ju g a te fa u l t p o p u la t io n A C -0 2 a l s o re tu rn e d s u rp r i s in g re s u l t s in c e R e c h e s ' m e th o d

1 3w a s to ta l ly u n a b le to de a l w i th th is se t o f fa u l t s ( f ig u re s 9 -4 1 th ro ug h 9 -4 5) . T h e F a n d F

o r ien ta t ion s w e re o n c e a g a in sw i tch e d a n d , m o re im p o r tan t ly , the va lue s fo r M ra n g e d f ro m

1 .0 0 to 6 .1 3 . T h e ra t io M , b y d e f in i t io n , ra n g e s fro m 0 .0 to 1 .0 o n ly . T h is im p l ie s th a t th e

1 3 1 3p ro gra m is a pp a re n t ly co nfu s in g th e F a n d F m a g n i tu de s w i th e a ch o th e r . W h e n th e F a n d F

or ien ta t ion s a re sw i tch ed , h ow ev er , t he so lu t ion i s c o r rec t . T h i s i s a rea sona b le re su l t s inc e m os t

2s tru c tu ra l g e o lo g y te x tb o o k s w o u ld g ra p h ic a l ly a s s ig n F to b e p a ra l le l t o th e in te rs e c t io n o f th e

c o n ju g a te se t (w h ic h i t i sn ' t in th is c a se ) .

A n g e l ie r 's m e th o d a ls o re tu rn e d w i ld ly in a c c u ra te re s u l t s fo r c o n ju g a te fa u l t p o p u la t io n

A C -02 ( f igu res 9 -46 th rou g h 9 -50 ) . A t lo w in i t i a l M v a lu e s (0 .0 0 , 0 .2 5 , a nd 0 .5 0 ) , th e p ro gra m

2 3c o n fuse d the o r ie n ta t ion s o f F a n d F a n d a t la rge r in i t ia l M v a lu e s (0 .7 5 a nd 1 .0 0 ) , th e p ro gra m

1 2c o n fuse d the o r ie n ta t ion s o f F a n d F . T h e e rro r in m a g ni tu de be tw e e n th e in i t ia l a nd ca lc u la te d

v a lue s fo r M ra n g e d fro m a lo w o f 0 .15 5 to a h ig h o f 0 .47 9 w h ic h is un a c c e p ta b le .

C o m p a r in g th e re s u l t s o f R e c h e s ' a n d A n g e l ie r 's m e th o d s fo r c o n ju g a te fa u l t p o p u la t io n

1 3A C -0 1 ( ta b le 9 -4 ) s ho w s th e 9 0° e r ro r in th e o r ie n ta t io ns o f th e F a n d F p r in c ip a l s t re ss a xe s

fo r R e c h e s ' m e th o d . T h e d e v ia t io n s in m a g n i tu d e b e tw e e n th e in i t ia l a n d c a lc u la te d v a lu e s fo r

M ra n g e d fro m a lo w o f 0 .01 9 to a h ig h o f 0 .64 8 .

C o m p a r in g th e re s u l t s o f R e c h e s ' a n d A n g e l ie r 's m e th o d s fo r c o n ju g a te fa u l t p o p u la t io n

1 2 3A C -0 2 ( ta b le 9 -5 ) s ho w s l a rg e d ev ia t io ns in th e o r ie n ta t io ns o f th e F , F , a n d F p r inc ipa l s t re s s

2a x e s fo r in i t i a l M v a lu e s o f 0 .0 0 , 0 .2 5 , a nd 0 .5 0 a nd la rg e d e v ia t io ns in th e o r ie n ta t io ns o f th e F

3a n d F p r inc ipa l s t re ss a x e s fo r in i t i a l M v a lu e s o f 0 .7 5 a n d 1 .0 0 . T h e d e v ia t io n s in m a g n i tu d e

b e tw e e n the in i t i a l a n d c a lcu la ted va lue s fo r M w e re l a rge (0 .5 2 1 ) fo r an in i t i a l M v a lu e o f 0 .0 0

a n d un d e f ine d for a l l o th e r in i t ia l M v a lu e s .

A g e n e ra l o b s e rv a t io n se em s to be th a t bo th A n g e lie r 's a n d R e ch e s ' m e th o d s o f

p a le o s t re ss a na ly s is re tu rn m o re re a so na b le re su lt s fo r c on ju ga te f a u l t s e t s w h ic h p a ra l le l

p r in c ip a l s t r e s s a x e s .

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.00 1.000 1.000

Figure 9-31 - Lower-hemisphere stereographic projection and table demonstrating the differences between

1 2 3the known and calculated principal stress axes F , F , and F and the value of M for fault population AC-01-

00 using Reches' method of paleostress analysis. In the stereographic projection, the known principal stress

1 2 3axes F , F , and F are oriented up, east, and north respectively and the calculated principal stress axes are

denoted by the filled circles labelled 1, 2, and 3.

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.25 0.695 0.305

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.50 0.485 0.515

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.75 0.274 0.726

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 1.00 0.089 0.911

Known Program Error

1F 90/000 90/051 0.0°

2F 00/090 00/269 1.0°

3F 00/000 00/179 1.0°

M 0.00 0.453 0.453

00 using Angelier's method of paleostress analysis. In the stereographic projection, the known principal stress

Known Program Error

1F 90/000 90/076 0.0°

2F 00/090 00/270 0.0°

3F 00/000 00/180 0.0°

M 0.25 0.477 0.227

Known Program Error

1F 90/000 90/050 0.0°

2F 00/090 00/270 0.0°

3F 00/000 00/180 0.0°

M 0.50 0.504 0.004

Known Program Error

1F 90/000 90/130 0.0°

2F 00/090 00/270 0.0°

3F 00/000 00/360 0.0°

M 0.75 0.531 0.219

Known Program Error

1F 90/000 90/102 0.0°

2F 00/090 00/270 0.0°

3F 00/000 00/360 0.0°

M 1.00 0.559 0.441

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.00 1.000 1.000

Known Program Error

1F 90/000 00/260 90.0°

2F 00/090 00/350 80.0°

3F 00/000 90/270 90.0°

M 0.25 1.358 ?

Known Program Error

1F 90/000 00/254 90.0°

2F 00/090 00/344 74.0°

3F 00/000 90/270 90.0°

M 0.50 2.732 ?

Known Program Error

1F 90/000 00/251 90.0°

2F 00/090 00/341 71.0°

3F 00/000 90/270 90.0°

M 0.75 6.133 ?

Known Program Error

1F 90/000 00/250 90.0°

2F 00/090 00/340 70.0°

3F 00/000 90/270 90.0°

M 1.00 1.312 ?

Known Program Error

1F 90/000 90/179 0.0°

2F 00/090 00/0 46.0°

3F 00/000 90/314 46.0°

M 0.00 0.479 0.479

Known Program Error

1F 90/000 90/192 0.0°

2F 00/090 00/060 30.0°

3F 00/000 00/330 30.0°

M 0.25 0.557 0.307

Known Program Error

1F 90/000 90/170 0.0°

2F 00/090 00/069 21.0°

3F 00/000 90/339 21.0°

M 0.50 0.772 0.272

Known Program Error

1F 90/000 00/073 90.0°

2F 00/090 90/184 90.0°

3F 00/000 00/343 17.0°

M 0.75 0.905 0.155

Known Program Error

1F 90/000 00/074 90.0°

2F 00/090 90/167 90.0°

3F 00/000 00/344 17.0°

M 1.00 0.676 0.324

1F 00/000 90/051 90.0°

2F 00/090 00/269 1.0°

3F 90/270 00/179 90.0°

M 1.000 0.453 0.547

1F 00/000 90/076 90.0°

2F 00/090 00/270 0.0°

3F 90/270 00/180 90.0°

M 0.695 0.477 0.218

1F 00/000 90/050 90.0°

2F 00/090 00/270 0.0°

3F 90/270 00/180 90.0°

M 0.485 0.504 0.019

1F 00/000 90/130 90.0°

2F 00/090 00/270 0.0°

3F 90/270 00/360 90.0°

M 0.274 0.531 0.257

1F 00/000 90/102 90.0°

2F 00/090 00/270 0.0°

3F 90/270 00/360 90.0°

M 0.089 0.559 0.648

1 2 3Table 9-4 - Table demonstrating the deviation for the orientations of the F , F , and F principal stress axes

and the principal stress magnitude ratio M between Reches' and Angelier's methods of paleostress analysis

on fault population AC-01 at five different values of M.

1F 00/000 90/179 90.0°

2F 00/090 00/044 46.0°

3F 90/270 00/314 90.0°

M 1.000 0.479 0.521

1F 00/260 90/192 90.0°

2F 00/350 00/060 70.0°

3F 90/270 00/330 90.0°

M 1.358 0.557 ?

1F 00/254 90/170 90.0°

2F 00/344 00/069 85.0°

3F 90/270 00/339 90.0°

M 2.372 0.772 ?

1F 00/251 00/073 2.0°

2F 00/341 90/184 90.0°

3F 90/270 00/343 90.0°

M 6.133 0.905 ?

1F 00/250 00/074 4.0°

2F 00/340 90/167 90.0°

3F 90/270 00/344 90.0°

M 1.312 0.676 ?

on fault population AC-02 at five different values of M.

T h e n e x t sp e c ia l -c a s e fa u l t p o p u la t io n to b e e x a m in e d w a s th e o r th o r h o m b ic s ym m e tr y

p o p u la t io n O S -0 1 . T h e o r th o rh o m b ic sym m e try fa u l t s c r e a te d a s i tu a t io n ve ry s im i la r to th e

c o n ju ga te fa u l t p o p u la t io n A C -0 1 . F o r R e c he s ' m e th o d ( f ig u re s 9 -5 1 t h ro u g h 9 -5 5 ) , th e M

v a lu e s r e tu rn e d , o nc e a g a in , se e m ed to b e th e o p p o s i te o f w ha t o ne w o u ld ex p ec t . A M o f 1 .0 0

w a s re tu r n e d fo r a n in i t i a l M o f 0 .0 0 , a M o f 0 .7 6 w a s re tu r n e d fo r a n in i t i a l M o f 0 .2 5 , a M o f

0 .5 1 w a s re tu r n e d fo r a n in i t i a l M o f 0 .5 0 (w h ic h i s q u i t e g o o d ) , a M o f 0 .2 6 w a s re tu rn e d fo r

a n in i t i a l M o f 0 .7 5 , a nd a M o f 0 .0 2 w a s r e tu r n e d fo r a n in i t i a l M o f 1 .0 0 . N o t sur p r i s in g ly ,

1 3th e F a n d F w e re o n c e a g a in sw itc he d a nd a c o r re c t so lu t io n re su l t s i f th e y a re c ha ng e d .

U s in g A n g e l ie r 's m e th o d ( f ig u re s 9 -5 6 th ro u g h 9 -6 0 ) , th e o r ie n ta t io n s o f th e p r in c ip a l

1 2 3s t r e s s ax es F , F , a n d F w e re e x a c t ly m a tch ed fo r in i t ia l M v a lue s o f 0 .2 5 , 0 .5 0 , a n d 0 .7 5 . F o r

2 3a n in i t i a l M v a lu e o f 0 .0 0 , ho w e v e r , the o r ie n ta t io n s o f F a n d F w e re re v e r se d a nd fo r an

1 2i n i t i a l M v a lu e o f 1 .0 0 , th e o r ie n ta t io n s o f F a n d F w e re r e ve rse d . T h e e r ro r s in m a g n i tu d e

fo r th e c a lcu la ted va lue o f M w e re a l l l e ss th a n 0 .0 5 0 fo r e a c h in i t i a l M va lue .

C o m p a r i n g t h e re su l t s o f R e c h e s ' to A n g e l ie r 's m e tho d s fo r o r th o rh o m b ic fau l t

p o p u la t io n O S -0 1 ( tab le 9 -6 ) sho w s la r ge d ev ia t io ns in the o r i en ta t io ns o f the p r inc ip a l s t r e s s

a x e s a n d in th e c a lc u la te d v a lu e s fo r M b u t th is i s p r im a r i ly d u e to th e in a c c u ra c y o f R e c h e 's

m e tho d g iv e n th i s typ e o f fa u l t p o p u la t io n .

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.00 1.000 1.000

1 2 3the known and calculated principal stress axes F , F , and F and the value of M for fault population OS-01-

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.25 0.755 0.505

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.50 0.502 0.002

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.75 0.261 0.489

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 1.00 0.015 0.985

Known Program Error

1F 90/000 90/196 0.0°

2F 00/090 00/360 90.0°

3F 00/000 00/090 90.0°

M 0.00 0.000 0.000

Known Program Error

1F 90/000 90/131 0.0°

2F 00/090 00/270 0.0°

3F 00/000 00/000 0.0°

M 0.25 0.266 0.016

Known Program Error

1F 90/000 90/126 0.0°

2F 00/090 00/270 0.0°

3F 00/000 00/000 0.0°

M 0.50 0.525 0.025

Known Program Error

1F 90/000 90/098 0.0°

2F 00/090 00/270 0.0°

3F 00/000 00/360 0.0°

M 0.75 0.776 0.026

Known Program Error

1F 90/000 00/270 90.0°

2F 00/090 90/090 90.0°

3F 00/000 00/180 0.0°

M 1.00 0.981 0.019

1F 00/000 90/196 90.0°

2F 00/090 00/360 90.0°

3F 90/270 00/090 90.0°

M 1.000 0.000 1.000

1F 00/000 90/131 90.0°

2F 00/090 00/270 0.0°

3F 90/270 00/000 90.0°

M 0.755 0.266 0.489

1F 00/000 90/126 90.0°

2F 00/090 00/270 0.0°

3F 90/270 00/000 90.0°

M 0.502 0.525 0.023

1F 00/000 90/098 90.0°

2F 00/090 00/270 0.0°

3F 90/270 00/000 90.0°

M 0.261 0.776 0.515

1F 00/000 00/270 90.0°

2F 00/090 90/090 0.0°

3F 90/270 00/180 90.0°

M 0.015 0.981 0.966

on fault population OS-01 at five different values of M.

T h e ra d ia l sym m e t ry fau l t p o p u la t io n R S -0 1 w a s a l so ve ry s im i la r to the c o n ju g a te fau l t

p o p u la t io n A C -0 1 . T h e M v a lu e s r e tu rn e d b y R e c h e s ' m e th o d ( f ig u re s 9 -6 1 th ro u g h 9 -6 5 ) , o n c e

a g a in , se e m ed to b e th e o p p o s i t e o f w h a t o ne w o u ld ex p ec t . A M o f 1 .0 0 w a s re tu rn e d fo r an

in i t i a l M o f 0 .0 0 , a M o f 0 .7 5 w a s re tu r n e d fo r a n in i t i a l M o f 0 .2 5 , a M o f 0 . 5 8 w a s re tu rn e d

fo r a n in i t i a l M o f 0 .5 0 (w h ic h i s q u i te go o d ) , a M o f 0 .3 9 w a s re tu r n e d fo r a n in i t i a l M o f 0 .7 5 ,

1 3a n d a M o f 0 .1 8 w a s re tu r n e d fo r a n in i t i a l M o f 1 .0 0 . N o t su rp r is in g ly , th e F a n d F w e re o n ce

a ga in sw itc he d a nd a c o r re c t so lu t io n re su l t s i f th e y a re c ha ng e d .

U s in g A n g e l ie r 's m e th o d ( f ig u re s 9 -6 6 th ro u g h 9 -7 0 ) , th e o r ie n ta t io n s o f th e p r in c ip a l

1 2 3s t r e s s ax es F , F , a n d F w e re e x a c t ly m a tch e d fo r in i t ia l M v a lu e s o f 0 .5 0 , 0 .7 5 , a n d 1 .0 0 . F o r

2 3in i t i a l M va lu e s o f 0 .0 0 a n d 0 .2 5 , th e o r ie n ta t io n s o f F a n d F w e r e s t i l l q u i te s m a l l . T h e

e r r o rs in m a g n i tud e fo r th e c a lc u la te d v a lu e o f M w e re a l l l e ss th a n 0 .0 5 0 fo r e a c h in i t i a l M

v a lue .

C o m p a r in g the re su l t s o f R e c h e s ' to A n g e l ie r 's m e tho d s fo r ra d ia l sym m e t ry fau l t

p o p u la t io n R S -0 1 ( ta b le 9 -7 ) sh o w s la rg e d e v ia t io n s in th e o r ie n ta t io n s o f th e p r in c ip a l s t r e s s

a x e s a n d in th e c a lc u la te d v a l u e s fo r M b u t th is i s p r im a r i ly d u e to th e in a c c u ra c y o f R e c h e 's

m e tho d g iv e n th i s typ e o f fa u l t p o p u la t io n .

I t se e m s th a t fo r c o n j u g a te - typ e f a u l t se t s ( th e o r th o r h o m b ic a n d th e r a d i a l sym m e tr y

fau l t p o p u la t io n s a re b o th typ e s o f c o n ju g a te fau l t se t s ) , R e c h e s ' m e tho d p a le o s t r e s s a n a lys i s

1 3p r o g ra m re v e r se d th e o r ie n ta t io n s o f th e F an d F a xe s a n d c a lc u la te d M v a lu e s b a se d o n th a t

s w i tc h . T h is s e e m s t o b e a r e su l t o f th e m a th e m a t ic a l a lg o r i th m s u s e d to c a lc u la te th e

p a leo s t r e s s ax es . O the rw ise , the p ro gra m s d o r e tu rn the co r rec t o r i en ta t io ns fo r the p a l eo s t r e s s

a x e s a n d w o u l d y ie ld a c o r re c t so lu t io n . T h e M v a lu e s r e tu rn e d l ik e wise a re o n ly so m e w ha t

in c o r re c t . T h e M va lue re tu r n e d fo r a n in i t i a l M o f 0 .5 0 i s e s sen t ia l ly 0 .5 0 . F o r R ec he s '

m e th o d , th e M v a lu e r e tu rn e d fo r 0 .2 5 is e sse n t ia l ly 0 .7 5 (a n d v ic e v e rsa ) , a nd th e M v a lu e

r e tu rn e d fo r 0 .0 i s e x a c t ly 1 .0 ( a n d v ic e v e rsa ) . F o r A n g e l ie r 's m e th o d , th e M v a lu e s r e tu rn e d

a re e sse n t ia l ly co r r e c t .

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.00 1.000 1.000

1 2 3the known and calculated principal stress axes F , F , and F and the value of M for fault population RS-01-

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.25 0.746 0.496

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.50 0.583 0.083

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 0.75 0.392 0.358

Known Program Error

1F 90/000 00/000 90.0°

2F 00/090 00/090 0.0°

3F 00/000 90/270 90.0°

M 1.00 0.184 0.816

Known Program Error

1F 90/000 90/102 0.0°

2F 00/090 00/263 7.0°

3F 00/000 00/353 7.0°

M 0.00 0.036 0.036

Known Program Error

1F 90/000 90/012 0.0°

2F 00/090 00/269 1.0°

3F 00/000 00/179 1.0°

M 0.25 0.252 0.002

Known Program Error

1F 90/000 90/081 0.0°

2F 00/090 00/270 0.0°

3F 00/000 00/180 0.0°

M 0.50 0.493 0.007

Known Program Error

1F 90/000 90/085 0.0°

2F 00/090 90/270 0.0°

3F 00/000 00/180 0.0°

M 0.75 0.742 0.008

Known Program Error

1F 90/000 00/090 0.0°

2F 00/090 90/270 0.0°

3F 00/000 00/180 0.0°

M 1.00 0.997 0.003

1F 00/000 90/102 90.0°

2F 00/090 00/263 7.0°

3F 90/270 00/353 90.0°

M 1.000 0.036 0.964

1F 00/000 90/012 90.0°

2F 00/090 00/269 1.0°

3F 90/270 00/179 90.0°

M 0.746 0.252 0.494

1F 00/000 90/081 90.0°

2F 00/090 00/270 0.0°

3F 90/270 00/180 90.0°

M 0.583 0.493 0.090

1F 00/000 90/085 90.0°

2F 00/090 00/270 0.0°

3F 90/270 00/180 90.0°

M 0.392 0.742 0.350

1F 00/000 00/090 90.0°

2F 00/090 90/270 90.0°

3F 90/270 00/180 90.0°

M 0.184 0.997 0.813

on fault population RS-01 at five different values of M.

F i n a l ly , t h e t h re e f a u l t p o p u la t i o n s S O -0 1 , S O -0 2 , a n d S O -0 3 c o n s i s t i n g o f f a u l t p la n e s

o f s im i l a r o r i e n ta t i o n s w e re t e s t e d . T h e s e fa u l t s a l s o g a v e s o m e w h a t u n e x p e c t e d r e s u l t s w h ic h

w e re a b i t d i f f e r e n t f ro m th e r e s u l t s g iv e n fo r t h e o th e r s p e c ia l - c a s e fa u l t p o p u la t i o n s t e s te d .

F i r s t , c o n s id e r th e r e s u l t s fo r R e c h e s ' m e th o d o n fa u l t p o p u la t i o n S O -0 1 ( f i g u re s 9 -7 1

2t h ro u g h 9 -7 5 ) . T h e o r i e n ta t i o n s fo r th e F p r i n c i p a l s t r e s s a x e s a re c o r re c t , w h ic h i s n o t to o

1 3s u rp r i s in g c o n s id e r in g th a t t h e f a u l t s a re a l l p a ra l l e l t o th e e a s t -w e s t a x i s , b u t th e F a n d F

a x e s a r e q u i t e a b i t o f f (± 1 8 ° ) . T h e M v a lu e s r e tu rn e d b y th e p ro g ra m a re a l s o v e ry o d d . T h e

M v a l u e r e t u r n e d f o r a n i n i t i a l M o f 0 .0 0 i s n e g a t i v e a n d th e M v a l u e s r e t u r n e d f o r i n i t i a l M

v a lu e s o f 0 .7 5 a n d 1 .0 0 a r e b o th g re a te r t h a n 1 .0 . S in c e M i s d e f in e d a s b e in g b e tw e e n 0 .0 a n d

1 .0 i n c l u s i v e , t h e s e v a l u e s a r e o b v i o u s l y i n e r ro r . In t h i s c a s e , h o w e v e r , s w i t c h i n g t h e a x e s w i l l

n o t h e lp .

2F o r A n g e l i e r ' s m e th o d , i n a s im i l a r f a s h io n , t h e o r i e n ta t i o n s fo r th e F p r in c ip a l s t r e s s

1 3a x e s a re e s s e n t i a l l y c o r re c t a n d th e F a n d F a x e s a r e q u i t e a b i t o f f (± 1 6 ° ) . T h e e r ro r s i n

m a g n i t u d e f o r t h e c a l c u l a t e d v a l u e s o f M w e r e l a r g e s t ( a l m o s t 0 . 4 ) a t a n i n i t i a l M v a lu e o f 0 .5 0

a n d s o m e w h a t s m a l l e r f o r l a r g e r a n d s m a l l e r i n i t i a l M v a lu e s .

C o m p a r i n g t h e r e s u l t s o f R e c h e s ' t o A n g e l i e r ' s m e t h o d s f o r s i m i l a r o r i e n t a t i o n f a u l t

p o p u la t i o n S O -0 1 ( t a b le 9 -8 ) s h o w s l a rg e d e v ia t i o n s (± 3 6 ° ) in th e o r i e n ta t i o n s o f th e p r in c ip a l

1 2 3s t r e s s a x e s F , F , a n d F f o r a n i n i t i a l M v a lu e o f 0 .0 0 b u t r a th e r s m a l l d e v ia t i o n s ( l e s s th a n

1 0 ° ) f o r a l l o f th e p r i n c i p a l s t re s s a x i s o r i e n t a t i o n s a t in i t i a l M v a lu e s o f 0 .2 5 , 0 .5 0 , 0 .7 5 , a n d

1 .0 0 .

Known Program Error

1F 90/000 66/180 24.0°

2F 00/090 00/090 0.0°

3F 00/000 24/000 24.0°

M 0.00 -0.047 ?

1 2 3the known and calculated principal stress axes F , F , and F and the value of M for fault population SO-01-

Known Program Error

1F 90/000 72/000 18.0°

2F 00/090 00/090 0.0°

3F 00/000 18/180 18.0°

M 0.25 0.581 0.331

Known Program Error

1F 90/000 72/000 18.0°

2F 00/090 00/090 0.0°

3F 00/000 18/180 18.0°

M 0.50 0.847 0.347

Known Program Error

1F 90/000 72/000 18.0°

2F 00/090 00/090 0.0°

3F 00/000 18/180 18.0°

M 0.75 1.125 ?

Known Program Error

1F 90/000 72/000 18.0°

2F 00/090 00/090 0.0°

3F 00/000 18/180 18.0°

M 1.00 1.364 1.000

Known Program Error

1F 90/000 77/002 13.0°

2F 00/090 01/268 2.2°

3F 00/000 13/178 13.2°

M 0.00 0.136 0.136

Known Program Error

1F 90/000 81/000 9.0°

2F 00/090 00/270 0.0°

3F 00/000 09/180 9.0°

M 0.25 0.334 0.084

Known Program Error

1F 90/000 76/002 14.0°

2F 00/090 01/270 1.0°

3F 00/000 14/180 14.0°

M 0.50 0.880 0.380

Known Program Error

1F 90/000 74/004 16.0°

2F 00/090 01/270 1.0°

3F 00/000 16/180 16.0°

M 0.75 0.939 0.189

Known Program Error

1F 90/000 74/007 16.0°

2F 00/090 02/270 2.0°

3F 00/000 16/180 16.0°

M 1.00 0.967 0.033

1F 66/180 77/002 37.0°

2F 00/090 01/268 2.2°

3F 24/000 13/178 37.1°

M -0.047 0.136 ?

1F 72/000 81/000 9.0°

2F 00/090 00/270 0.0°

3F 18/180 09/180 9.0°

M 0.581 0.334 0.218

1F 72/000 76/002 4.0°

2F 00/090 01/270 1.0°

3F 18/180 14/180 4.0°

M 0.847 0.880 0.019

1F 72/000 74/004 2.3°

2F 00/090 01/270 1.0°

3F 18/180 16/000 2.0°

M 1.125 0.939 ?

1F 72/000 74/007 2.9°

2F 00/090 02/270 2.0°

3F 18/180 16/180 2.0°

M 1.364 0.967 ?

on fault population SO-01 at five different values of M.

T h e r e s u l t s f ro m R e c h e s ' m e th o d ( f i g u re s 9 -8 1 th ro u g h 9 -8 5 ) a n d A n g e l i e r ' s m e th o d

( f ig u re s 9 -8 6 th ro u g h 9 -9 0 ) fo r s im i l a r o r i e n ta t i o n f a u l t p o p u la t i o n S O -0 2 a r e e v e n w o rs e th a n

th o s e fo r S O -0 1 . N o n e o f t h e t h re e p r in c ip a l s t r e s s a x e s a re c o r re c t a n d n o n e o f th e M v a lu e s

a r e e v e n c l o s e t o b e i n g w h a t th e y s h o u l d b e . C o m p a r i n g R e c h e s ' t o A n g e l i e r ' s m e t h o d ( ta b l e

9 -9 ) s h o w s th a t , w h i l e b o th in c o r r e c t , th e y a re a l s o b o th c o n s i s te n t .

T h e r e s u l t s f ro m R e c h e s ' m e th o d ( f i g u re s 9 -9 0 th ro u g h 9 -9 5 ) a n d A n g e l i e r ' s m e th o d

( f ig u re s 9 -9 6 t h r o u g h 9 - 1 0 0 ) fo r s im i l a r o r i e n ta t i o n fa u l t p o p u la t i o n S O -0 3 h a s a s im i l a r

p ro b le m . N o n e o f th e t h re e p r in c ip a l s t r e s s a x e s a re c o r re c t a n d n o n e o f th e M v a lu e s a r e e v e n

c lo s e to b e in g w h a t th e y s h o u ld b e . C o m p a r in g R e c h e s ' t o A n g e l i e r ' s m e th o d ( t a b l e 9 -1 0 )

s h o w s th a t t h e s e re s u l t s a re a l s o c o n s i s te n t .

I a m n o t a b l e t o g iv e a s a t i s fa c t o ry e x p l a n a t io n o f w h y th e s e tw o f a u l t p o p u la t i o n s

s h o u l d b e s o f a r o f f . I t i s p r o b a b l y d u e t o t h e f a c t th a t w h e n y o u h a v e a p o p u l a t i o n o f f a u l t s

w h e re a l l o f th e m a re o f a s im i l a r o r i e n t a t i o n , t h e re i s n o t e n o u g h o f a c o n s t r a in t u p o n th e

lo c a t io n o f th e p r in c ip a l s t r e s s a x e s . T h e r a t i o n a le b e h in d a n e g a t iv e M v a lu e b e in g re tu rn e d

fo r S O -0 2 w h e n M i s i n i t i a l l y 0 .0 0 i s a m y s t e r y . I w a s n o t a b l e to a s c e r t a in w h y n e g a t iv e M

v a lu e s sh o u ld b e re tu rn e d . T h e r e a s o n i s u n d o u b ta b ly c o n ta in e d w i th in th e s o u rc e c o d e o f th e

p ro g ra m s a n d s im p le e r ro r c h e c k i n g s h o u ld h a v e b e e n a b l e t o c o n s t r a in M t o b e b e tw e e n 0 .0 a n d

1 .0 in c lu s iv e . A v a lu e o u ts id e o f t h a t r a n g e s h o u ld n o t b e a l l o w e d .

Known Program Error

1F 90/000 66/225 24.0°

2F 00/090 24/045 49.8°

3F 00/000 00/315 45.0°

M 0.00 0.045 0.045

Known Program Error

1F 90/000 81/175 9.0°

2F 00/090 01/272 2.2°

3F 00/000 09/002 9.2°

M 0.25 0.110 0.140

Known Program Error

1F 90/000 70/148 20.0°

2F 00/090 15/287 22.5°

3F 00/000 13/020 23.7°

M 0.50 0.275 0.225

Known Program Error

1F 90/000 55/135 35.0°

2F 00/090 30/282 32.1°

3F 00/000 16/021 26.2°

M 0.75 0.464 0.286

Known Program Error

1F 90/000 34/123 56.0°

2F 00/090 52/270 52.0°

3F 00/000 17/022 27.5°

M 1.00 0.469 0.531

Known Program Error

1F 90/000 7/047 13.0°

2F 00/090 01/313 43.0°

3F 00/000 13/223 44.6°

M 0.00 0.130 0.130

Known Program Error

1F 90/000 77/051 13.0°

2F 00/090 07/285 16.5°

3F 00/000 10/194 17.1°

M 0.25 0.252 0.002

Known Program Error

1F 90/000 75/082 15.0°

2F 00/090 14/284 19.7°

3F 00/000 05/192 13.0°

M 0.50 0.615 0.015

Known Program Error

1F 90/000 61/103 29.0°

2F 00/090 29/279 30.2°

3F 00/000 01/010 10.0°

M 0.75 0.746 0.004

Known Program Error

1F 90/000 37/101 53.0°

2F 00/090 52/270 52.0°

3F 00/000 05/0070 8.6°

M 1.00 0.786 0.214

Known Program Error

1F 90/000 49/210 41.0°

2F 00/090 41/030 67.8°

3F 00/000 00/120 60.0°

M 0.00 0.202 0.202

Known Program Error

1F 90/000 59/203 31.0°

2F 00/090 30/036 59.4°

3F 00/000 06/303 57.2°

M 0.25 0.191 0.059

Known Program Error

1F 90/000 59/191 31.0°

2F 00/090 20/065 31.6°

3F 00/000 23/326 40.3°

M 0.50 0.082 0.418

Known Program Error

1F 90/000 57/175 33.0°

2F 00/090 05/273 5.8°

3F 00/000 33/007 33.7°

M 0.75 0.207 0.543

Known Program Error

1F 90/000 50/155 40.0°

2F 00/090 19/270 19.0°

3F 00/000 33/013 35.2°

M 1.00 0.408 0.592

Known Program Error

1F 90/000 80/202 10.0°

2F 00/090 02/302 32.1°

3F 00/000 10/032 33.4°

M 0.00 0.000 0.000

Known Program Error

1F 90/000 79/189 11.0°

2F 00/090 02/287 17.1°

3F 00/000 11/017 20.2°

M 0.25 0.190 0.060

Known Program Error

1F 90/000 79/184 11.0°

2F 00/090 01/090 1.0°

3F 00/000 11/360 11.0°

M 0.50 0.299 0.201

Known Program Error

1F 90/000 79/187 11.0°

2F 00/090 01/090 1.0°

3F 00/000 11/359 11.0°

M 0.75 0.584 0.166

Known Program Error

1F 90/000 69/130 21.0°

2F 00/090 16/270 16.0°

3F 00/000 13/004 13.6°

M 1.00 0.865 0.135

1F 66/225 77/047 37.0°

2F 24/045 01/313 88.6°

3F 00/315 13/223 88.1°

M 0.045 0.130 0.085

1F 81/175 77/051 19.5°

2F 01/272 07/285 14.3°

3F 09/002 10/194 22.4°

M 0.110 0.252 0.142

1F 70/148 75/082 19.3°

2F 15/287 14/284 3.1°

3F 13/020 05/192 19.7°

M 0.275 0.615 0.340

1F 55/135 61/103 17.8°

2F 30/282 29/279 2.8°

3F 16/021 01/010 18.5°

M 0.464 0.746 0.282

1F 34/123 37/101 18.1°

2F 52/270 52/270 0.0°

3F 17/022 05/007 19.0°

M 0.469 0.786 0.317

1F 49/210 80/202 36.1°

2F 41/030 02/302 87.2°

3F 00/120 10/032 88.0°

M 0.202 0.000 0.202

1F 59/203 79/189 20.5°

2F 30/036 02/287 74.7°

3F 06/303 11/017 73.2°

M 0.191 0.190 0.001

1F 59/191 79/184 20.1°

2F 20/065 01/090 31.0°

3F 23/326 11/360 34.5°

M 0.082 0.299 0.217

1F 57/175 79/187 22.3°

2F 05/273 01/090 6.7°

3F 33/007 11/359 23.2°

M 0.202 0.584 0.382

1F 50/155 69/130 22.5°

2F 19/270 16/270 3.0°

3F 33/013 13/004 21.6°

M 0.408 0.865 0.457

9 .4 O t h e r T e s t R e s u l t s

T h e p a l e o s t r e s s a n a ly s i s p ro g ra m s w e re n e x t c h e c k e d f o r th e i r s e n s i t i v i t y to a

v a r i a b i l i t y i n th e o r ie n t a t io n s o f t h e f a u l t p l a n e s a n d in t h e p i t c h a n g l e s o f th e s l i p v e c t o r s .

T h i s w a s d o n e b y a rb i t r a r i l y c h a n g in g th e o r i e n t a t i o n s o f o n e o r m o re o f th e f a u l t d a tu m s a n d

th e n r e c a l c u la t i n g th e p a l e o s t r e s s t e n s o r fo r th a t p o p u l a t i o n . T h e s e t e s t s w e re d o n e w i th th e

R P -0 1 , R P -0 2 , a n d R P -0 3 fa u l t p o p u la t i o n s . T h e p re l im in a ry r e s u l t s i n d i c a te th a t t h e p ro g ra m s

a re n o t v e ry s e n s i t i v e t o m i n o r c h a n g e s (± 5 ° ) in e i t h e r th e s t r i k e , d ip , o r p i t c h a n g l e s o f th e

fa u l t d a ta .

T h e p a l e o s t r e s s a n a l y s i s p r o g r a m s w e r e a l s o c h e c k e d f o r t h e i r s e n s i t i v i t y f o r ra n d o m l y

in s e r t i n g o r r e m o v in g a f a u l t p l a n e f ro m th e p o p u la t i o n . T h e s e t e s t s w e re p e r fo rm e d o n a l l o f

t h e f a u l t p o p u l a t i o n s d i s c u s s e d . A d d i n g o r r e m o v i n g a f a u l t p l a n e f ro m a p o p u l a t i o n u s u a l l y

h a d l i t t l e e f f e c t e x c e p t in c a s e s w h e re th e f a u l t p la n e s w e re p a ra l l e l t o o n e o f th e p r in c ip a l

p l a n e s o r i f t h e y w e r e t h e o n l y f a u l t p l a n e s a t s o m e s p e c i f ic o r i e n t a t i o n w h i c h w a s s u f f i c i e n t l y

fa r f r o m th e o th e r fa u l t p la n e s .

I n g e n e r a l , t h e t w o p a l e o s t r e s s a n a l y s i s p r o g r a m s t e s t e d r e t u r n e d c o n s i s t e n t re s u l t s

( w i t h s o m e n o t a b l e e x c e p t i o n s ) . T h i s i s im p o r t a n t s i n c e i f t w o s e p a r a t e p r o g r a m s r e t u r n w i d e l y

d i f fe r i n g re s u l t s , th e n o n e o r b o th o f t h e m a re w ro n g .

Known Program Error

1F ? 84/255 ?

2F ? 06/068 ?

3F ? 01/158 ?

M ? 0.050 ?

Figure 9-101 - Lower-hemisphere stereographic projection and table showing the calculated principal stress

1 2 3axes F , F , and F and the value of M for fault population FD-01-00 using Reches' method of paleostress

analysis. [38 Neogene-age normal faults from central Crete (Angelier, 1979); see p. 173, and Table 8-1, p. 175]

Known Program Error

1F ? 79/237 ?

2F ? 11/053 ?

3F ? 01/143 ?

M ? 0.225 ?

Figure 9-102 - Lower-hemisphere stereographic projection and table showing the calculated principal stress

1 2 3axes F , F , and F and the value of M for fault population FD-01-00 using Angelier's method of paleostress

analysis. [38 Neogene-age normal faults from central Crete (Angelier, 1979); see p. 173, and Table 8-1, p. 175]

C H A P T E R 1 0

C O N C L U S IO N S

N o w th a t th e r a t i o n a l e fo r th e t e s t s , t h e t e s t i n g p ro c e d u r e s , a n d th e t e s t r e s u l t s h a v e

b e e n e x p la in e d , i t i s t i m e to p u t i t a l l to g e th e r .

1 0 .1 W h a t d o th e R e su lt s M e a n ?

O n e m a y d ra w s e v e ra l c o n c lu s io n s f ro m th e d a ta p re s e n te d in c h a p te r 9 .

1 . T h e t w o m e t h o d s o f p a l e o s t r e s s a n a l y s i s e x a m i n e d s e e m t o w o r k f a i r ly w e l l f o r f a u l t

p o p u la t i o n s w i th m o d e ra te ly - s c a t t e re d fa u l t s s u c h a s t h o s e in p o p u la t i o n s R P -0 1 , R P -

0 2 , a n d R P - 0 3 . A p o p u l a t i o n o f f a u l t s w i th s o m e s c a t t e r e v id e n t ly a c t s t o c o n s t r a in th e

p o s s ib l e p o s i t i o n s o f th e p r in c ip a l s t re s s a x e s . W i th s u c h f a u l t p o p u la t i o n s , t h e

p ro g ra m s s e e m e d to r e tu rn b e t t e r s t r e s s a x e s o r i e n ta t i o n s a n d b e t t e r c a l c u la t e d v a lu e s

o f M f o r c a s e s w h e re M w a s n o t e q u a l to 0 . 0 o r 1 .0 . T h i s i s re a s o n a b l e s i n c e i t i s

im p o s s ib l e t o d i s t i n g u i s h b e tw e e n tw o o f th e s t r e s s a x e s w h e n th e y h a v e th e s a m e

2 3 2 1m a g n i tu d e s ( i . e . a t M = 0 .0 , F = F a n d a t M = 1 .0 , F = F ) .

2 . T h e tw o m e th o d s o f p a l e o s t r e s s a n a ly s i s e x a m in e d s e e m e d n o t to w o rk v e ry w e l l fo r

s p e c ia l - c a s e fa u l t p o p u la t i o n s . T h i s i s m o s t - l i k e ly d u e t o t h e f a c t t h a t s u c h p o p u la t i o n s

c a n n o t w e l l - c o n s t ra i n t h e p a l e o s t r e s s a x e s ( c h a p t e r 2 ) . W h i le s u c h s p e c i a l -c a s e

p o p u l a t i o n s w o u l d p r o b a b l y n e v e r b e u s e d f o r a p a l e o s t r e s s a n a l y s i s , m o r e w o r k i s

n e e d e d to d e t e rm in e i f c o n ju g a t e o r o r th o rh o m b ic f a u l t s e t s w i th i n m o d e ra t e ly -

s c a t t e re d fa u l t p o p u la t i o n s w o u ld h a v e a n a d v e rs e e f f e c t .

3 . T h e t w o m e t h o d s o f p a l e o s t r e s s a n a l y s i s e x a m i n e d s e e m e d t o b e f a i r l y i m m u n e t o m i l d

m e a s u r e m e n t e r r o r s a n d a s m a l l a m o u n t o f e x t r a n e o u s d a t a . M u c h o f t h e s e n s i t i v i t y i s

a r e s u l t o f e x a c t l y w h i c h p l a n e s a r e r e m o v e d o r a d d e d a n d m o r e w o r k i s n e e d e d t o

q u a n t i fy th i s . T h i s i s n e e d e d in o rd e r fo r p e o p le w h o u s e p a l e o s t r e s s a n a ly s i s p ro g ra m s

to re a l i z e w h a t ty p e o f a c c u ra c y th e i r r e s u l t s h a v e . T h e c a l c u la t i o n o f e r ro r c o n e s

a r o u n d th e r e t u r n e d p r in c ip a l s t r e s s a x e s w o u ld b e a u s e fu l a d d i t i o n t o p a l e o s t r e s s

p ro g ra m s .

4 . F i n a l ly , t h e tw o p a l e o s t r e s s a n a ly s i s p ro g ra m s e x a m in e d s e e m e d to c o m p a r e t o o n e

a n o th e r f a i r l y w e l l . T h e r e s u l t s r e t u rn e d fo r m o s t p o p u la t i o n s w e re c lo s e e n o u g h fo r

e i t h e r p r o g r a m t o b e u s e d w i t h a p p r o x i m a t e l y e q u a l a c c u r a c y . T h e e x c e p t i o n s t o t h i s

w e r e w h e n o n e o r t h e o t h e r o f t h e p r o g r a m s " b l e w u p " ( a s i n t h e c a s e o f c o n j u g a t e f a u l t

s e t s ) a n d r e t u r n e d t o t a l l y i n c o r re c t d a t a . M o r e w o r k i s n e e d e d t o s e e i f t h e s e f a v o r a b l e

c o m p a r i s o n s h o ld u p w i t h a d d i t io n a l t e s t s .

I n m o s t r e g a r d s , R e c h e s ' a n d A n g e l i e r ' s m e t h o d s o f p a l e o s t r e s s a n a l y s i s s e e m e d t o

p e r fo r m f a i r l y w e l l . C a u t i o n m u s t b e u s e d w h e n a p p l y i n g t h e s e m e t h o d s , h o w e v e r , a n d m o r e

s y s t e m a t i c t e s t s a re n e e d e d to fu r th e r o u t l i n e t h e tw o p ro g ra m 's l im i t a t i o n s fo r c e r t a in ty p e s

o f fa u l t p o p u l a t io n s .

1 0 .2 P r a c t ic a l P r o b le m s in E v a lu a t in g P a le o str e ss A n a ly s is P r o g r a m s

T h e r e a r e s e v e r a l i m p o r t a n t p r a c t i c a l p r o b l e m s w h i c h m a y a r i s e w h e n a t t e m p t i n g t o

c o m p a re a n d e v a lu a te p a le o s t r e s s a n a ly s i s p ro g ra m s . A fe w o f t h e m a re l i s t e d b e lo w .

1 . W h e n u s i n g a p r o g r a m w r i t t e n b y s o m e o n e e l s e f o r p e r fo r m i n g p a l e o s t r e s s a n a l y s i s

t e c h n i q u e s , o n e m u s t b e s u r e th a t t h e p r o g r a m d o e s w h a t i t 's s u p p o s e d t o d o . I t i s v i r t u a l l y

im p o s s ib le to t e s t a p ro g ra m s u c h th a t o n e w i l l h a v e 1 0 0 % c o n f id e n c e in i t s p e r fo rm a n c e .

T h e r e f o r e , i t i s u s u a l l y a d v i s a b l e t o c a r e f u l l y c h e c k t h e p r o g r a m 's s o u r c e c o d e f o r a n y p o s s i b l e

e r ro rs . T h i s m a y b e a n e x t r e m e ly d i f f i c u l t a n d t im e -c o n s u m in g p ro je c t s i n c e p ro g ra m m e rs m a y

n o t w r i t e th e i r s o u rc e c o d e i n a c l e a r a n d c o n s i s t e n t m a n n e r . P o o r ly d o c u m e n te d p ro g ra m s m a y

b e a l m o s t i m p o s s i b l e t o r e a d a n d u n d e r s t a n d s i n c e t h e s o u r c e c o d e f o r p a l e o s t r e s s a n a l y s i s

p ro g ra m s m a y e a s i l y ru n in to h u n d re d s o f p r in t e d p a g e s . T h e re i s a l s o th e p ro b le m o f

p r o g r a m m e r s w h o w i l l n o t a l lo w y o u to s e e th e i r s o u r c e c o d e ( s u c h a s A n g e l i e r ) . O n e m u s t

th e n m a k e a d e c i s io n a s to w h e th e r o r n o t t o u s e t h e i r p ro g ra m . S i n c e p u b l i s h e d f i e ld s tu d i e s

h a v e m a d e u s e o f A n g e l i e r ' s m e th o d " a s i s " , I c h o s e t o t e s t i t e v e n th o u g h I h a d n o a c c e s s t o th e

o r ig in a l s o u rc e c o d e . A p o s s ib l e w a y a ro u n d t h i s p ro b l e m m ig h t b e to w r i t e y o u r o w n

p a le o s t r e s s a n a ly s i s p ro g ra m s ba se d up o n th e i r pu b l i sh e d m a th e m a t i c a l a lg o r i t h m s . S in c e s u c h

p ro g ra m s w o u ld o f t e n u s e s im i l a r p ro c e d u r e s , a m o d u la r a n d h ig h ly - s t ru c tu re d p ro g ra m m in g

la n g u a g e s u c h a s C w o u ld b e p re f e r a b l e t o m o s t o th e r s s i n c e th a t w o u ld a l l o w th e s h a r in g o f

m a n y s u b ro u t in e s b y s e v e ra l d i f f e re n t p ro g ra m s . A d ra w b a c k o f t h i s s o lu t io n , h o w e v e r , i s t h a t

i t w o u ld b e v e ry t im e -c o n s u m in g a n d a l s o s o m e w h a t f ru s t r a t i n g s in c e p u b l i s h e d d e sc r ip t i o n s

o f th e m a th e m a t i c a l a lg o r i t h m s u s e d b y p a l e o s t r e s s a n a ly s i s p ro g ra m s a r e n o t a lw a y s c le a r a n d

e a s y to fo l l o w . I t i s o b v io u s ly a l e s s - a p p e a l i n g p r o sp e c t th a n o b ta in in g a w o rk in g (a n d

p re s u m a b ly te s te d ) p ro g ra m f ro m s o m e o n e e l s e .

2 . C lo se ly r e l a t e d to t h e a b o v e p ro b le m i s t h a t p a l e o s t r e s s a n a ly s i s p ro g ra m s a re w r i t t e n

in m a n y d i f f e r e n t c o m p u te r l a n g u a g e s fo r s e v e r a l d i f f e r e n t t y p e s o f c o m p u te r s . I h a v e

p ro g ra m s w r i t t e n in F O R T R A N f o r a n I B M P C , B A S I C fo r a n IB M P C , F O R T R A N fo r

a M a c in to sh I I , F O R T R A N fo r a V A X -8 6 5 0 m a in f ra m e , a n d a

p a le o s t r e s s a n a ly s i s p ro g ra m c a l l e d R O M S A f ro m L i s l e (1 9 8 8 ) in a n o d d v e r s io n o f B A S IC th a t

d o e s n o t s e e m to ru n c o r re c t ly o n a n y c o m p u te r I 'v e u se d . W h e n c o m p a r in g p ro g ra m s w r i t t e n

in d i f f e r e n t l a n g u a g e s fo r d i f f e r e n t m a c h i n e s , i t i s d i f f i c u l t t o d e t e rm in e i f a n e r ro r i s d u e to th e

m a th e m a t i c a l a lg o r i t h m u s e d , th e l a n g u a g e u s e d , o r th e m a c h i n e u s e d . D i f f e r e n t l a n g u a g e s a n d

m a c h i n e s w i l l h a v e th e i r o w n w a y s o f t ru n c a t in g n u m b e r s , ro u n d in g -o f f n u m b e r s , a n d e r ro r

h a n d l i n g .

3 . W h e n a p a l e o s t r e s s a n a l y s i s p ro g ra m re tu rn s a n i n c o r re c t r e s u l t , i t i s d i f f i c u l t t o

e v a l u a t e t h e s o u r c e o f t h a t e r r o r . I s t h e e r r o r a m e c h a n i c a l e r r o r ( a p l u s s i g n w h e r e

th e re s h o u ld b e a m i n u s s i g n in th e p ro g ra m ) o r th e r e s u l t o f a n in h e re n t f l a w in th e

t e c h n i q u e ? D id th e e r ro r a r i s e a s a re s u l t o f th e m a th e m a t i c a l t e c h n i q u e s u se d o r

b e c a u s e t h e i n i t i a l a s s u m p t i o n s u n d e r l y i n g t h e w h o l e c o n c e p t a r e i n c o r re c t . T h e s e a r e

v e ry d i f f i c u l t p ro b le m s to a d d re s s a n d o n l y b y th o ro u g h t e s t i n g c a n th e y b e e v a l u a t e d .

4 . A n o th e r p r o b l e m fa c e d w h e n a t t e m p t in g to c o m p a re v a r io u s m e t h o d s o f p a l e o s t r e s s

a n a ly s i s i s t h a t m o s t o f th e m , b a s e d u p o n p e r s o n a l e x p e r i e n c e , r e q u i r e th e i r i n i t i a l

f a u l t - s l i p d a t a t o b e fo rm a t t e d i n d i f f e r e n t w a y s . S h o u l d th e f a u l t o r i e n t a t i o n d a t a b e

e n t e r e d a s a s t r i k e , d ip , a n d d ip d i r e c t i o n fo r e a c h f a u l t p l a n e o r a s a s t r i k e a n d d ip fo r

e a c h fa u l t p l a n e w h e re t h e s t r i k e i s a s su m e d to fo l l o w a r i g h t -h a n d ru l e ( t h e s t r i k e i s t h e

t r e n d o f t h e f a u l t 's p o l e + 9 0 ° ) o r s i m p l y a s a p l u n g e a n d t r e n d f o r t h e p o l e t o e a c h f a u l t

p la n e ? S h o u ld th e o r i e n ta t i o n s o f th e s t r i a t i o n s o n th e f a u l t p la n e b e e n te re d a s p i t c h

a n g l e s o r a s p lu n g e a n d t r e n d v a l u e s o r s im p ly a s a t r e n d ( s in c e t h e p lu n g e w i l l b e

c o n s t r a in e d b y th e f a u l t ' s o r i e n t a t i o n )? S h o u ld t h e f a u l t ' s s e n s e o f s l i p b e g iv e n a s u p ,

d o w n , l e f t , o r r i g h t o r s h o u l d i t b e s p e c i f i e d b y th e p i t c h a n g l e o f th e s t r i a t i o n s o r

s h o u ld i t b e g i v e n a s a c lo c k w i s e o r c o u n te r c l o c k w is e r o t a t io n a b o u t s o m e a x i s ?

F i n a l ly , s h o u ld th e f a u l t d a t a b e a s s ig n e d s o m e ty p e o f w e ig h t in g s c h e m e ?

E v e ry o n e w r i t i n g p a l e o s t r e s s a n a ly s i s p ro g ra m s h a s th e i r o w n p re f e r e n c e s a n d , u n fo r tu n a t e ly ,

t h e y n e v e r s e e m t o c o r re s p o n d . A w a y a r o u n d t h i s m e s s w o u l d b e t o w r i te a l l o f th e a n a l y s i s

p ro g ra m s y o u r s e l f a n d u s e a c o n s i s te n t s c h e m e fo r e n t e r in g th e d a t a - - t h i s o p t io n h a s t h e

d ra w b a c k s l i s te d i n n u m b e r 1 a b o v e . A n o th e r p o s s ib l e s o lu t io n i s t o w r i t e a p ro g ra m to d o th e

c o n v e rs io n s a n d th e n w r i t e t h e c o n v e r t e d d a ta t o a n i n p u t f i l e fo r e a c h ty p e o f a n a ly s i s p ro g ra m

u s e d . S u c h a p r o g r a m w o u l d h a v e t h e a d v a n t a g e o f r e c e i v i n g d a t a i n a n y f o r m a t y o u w e r e

c o m fo r t a b l e w i th a n d a u to m a t i c a l l y c o n v e r t i n g i t fo r y o u . D o in g a l l o f th e c o n v e r s io n s b y h a n d

f o r s e v e r a l d i f fe r e n t p a l e o s t r e s s a n a l y s i s m e t h o d s c a n b e e x t r e m e l y t i m e - c o n s u m i n g a n d i s

o b v io u s ly m o re p ro n e to r a n d o m e r ro r s th a n i s a n a u to m a te d c o n v e r s io n s y s t e m .

5 . U s in g p ro g r a m s w r i t t e n b y s o m e o n e e l s e h a s o th e r p ro b le m s a s w e l l . I 'v e r e c e iv e d

p a l e o s t r e s s a n a ly s i s p ro g ra m s w i t h s e v e ra l p a g e s o f f a i r l y c l e a r d o c u m e n ta t io n , I 'v e

re c e iv e d p ro g ra m s w i th se v e ra l p a g e s o f d o c u m e n ta t io n in F re n c h , a n d I 'v e r e c e iv e d

p r o g r a m s w i th a b s o lu t e l y n o d o c u m e n ta t io n w h a t s o e v e r . A t t e m p t in g to u s e

u n d o c u m e n te d p ro g ra m s i s a " t r i a l a n d e r ro r" p ro c e d u re w h ic h in v o lv e s m u c h w a s t e d

t im e a n d e f fo r t . A n g e l i e r ' s p r o g r a m (c h a p t e r 6 ) i s p ro b a b ly th e w o rs t i n th a t r e g a r d .

I fo u n d h i s p ro g ra m to b e e x c e e d in g ly d i f f i c u l t t o u s e - - e v e n a f t e r h a v in g e x p e r i e n c e

w i th se v e ra l o th e r m e th o d s o f p a l e o s t r e s s a n a ly s i s . I t w a s a n u n d o c u m e n te d , c o m p i l e d

p r o g r a m w i t h u n c l e a r a n d m i s l e a d i n g i n t e r a c t i v e p r o m p t s a n d I f o u n d i t t o b e v e r y

c o u n t e r - in tu i t i v e i n th e w a y i t w o rk e d . T h e p ro g ra m u s in g R e c h e s ' m e th o d o f

p a le o s t r e s s a n a ly s i s w h ic h w a s w r i t t e n b y K e n n e th H a rd c a s t l e ( c h a p te r 7 ) , o n th e o th e r

h a n d , w a s v e ry e a s y t o u s e . M o s t p a le o s t r e s s a n a ly s i s p ro g ra m s l i e b e tw e e n th o s e tw o

e x t r e m e s .

6 . S o m e m e th o d s o f p a l e o s t re ss a n a ly s is a re v e ry c o m p u ta t io n a l ly - in t e n s iv e . G e p h a r t a n d

F o r sy th 's m e th o d (1 9 8 4 ) w a s n o t e x a m in e d in m y s tu d y fo r t h e s im p le r e a s o n th a t a p o p u la t i o n

o f 2 0 f a u l t s t a k e s a fu l l 2 4 -h o u r d a y to ru n ! T e s t i n g d o z e n s o f s u c h fa u l t s ( e s p e c i a l l y w i th o u t

o w n i n g a n a p p r o p r ia t e c o m p u te r ) i s a m a j o r p r o j e c t . T e s t in g n o n - l i n e a r in v e r s io n m e t h o d s ,

s u c h a s A n g e l i e r ' s , m a y a l s o b e v e r y t i m e - i n t e n s i v e f o r l a r g e f a u l t p o p u l a t i o n s . A s f a u l t s a r e

a d d e d to th e p o p u la t i o n s a r i t h m a t i c a l l y , t h e t im e n e e d e d to pe r fo rm a n a n a ly s i s in c re a se s

e x p o n e n t i a l l y d u e to th e m a th e m a t ic s in v o lv e d in p e r fo rm in g m a t r ix in v e r s io n s (w h e re t h e

m a t r ic e s m a y h a v e a ra n k o f 2 n w h e re n i s th e n u m b e r o f fa u l t s in th e p o p u la t i o n e x a m in e d ) .

7 . S i n c e m o s t o f t h e p io n e e r s i n p a l e o s t r e s s a n a l y s i s a r e F r e n c h , m a n y o f t h e i m p o r t a n t

p a p e r s a n d e v e n s o m e o f t h e p ro g ra m d o c u m e n ta t io n i s w r i t t e n in th a t l a n g u a g e . I f o n e

d o e s n o t p o s s e s s a r e a d i n g k n o w le d g e o f F r e n c h , i t i s d i f f i c u l t t o l e a r n a l l t h a t o n e

s h o u l d a b o u t p a l e o s t r e s s a n a l y s i s t e c h n i q u e s . I w a s f o r t u n a t e i n h a v i n g a c c e s s t o

s o m e o n e (D e b ra L e n a rd o f S U N Y A lb a n y ) w h o w a s a b l e to t r a n s l a t e F re n c h te c h n ic a l

p a p e r s i n to E n g l i s h w i th o u t t o o m u c h d i f f i c u l ty . T h e p ro c e s s d id , h o w e v e r , t a k e s o m e

t i m e a n d e f f o r t .

8 . F i n a l ly , i t i s v e ry d i f f i c u l t t o c h o o s e a r t i f i c i a l f a u l t p o p u l a t i o n s fo r t e s t i n g th e v a r io u s

m e th o d s o f p a l e o s t r e s s a n a ly s i s . S i n c e a n i n f in i t e v a r i e ty o f p o p u l a t i o n s m a y b e

c r e a t e d , c a re f u l t h o u g h t m u s t g o i n t o th e p r o b l e m . O b v io u s ly , i t i s d e s i ra b l e t o t e s t

s p e c i a l -c a s e s (e .g . a p o p u la t i o n c o n ta in in g a f a u l t p la n e p a ra l l e l t o o n e o f th e p r in c ip a l

s t r e s s a x e s ) a n d g e o lo g ic a l ly - re a l i s t i c f a u l t p o p u l a t io n s (e .g . o r t h o r h o m b i c s y m m e t r y

fa u l t s e t s ) . E v e ry p o s s ib l e t y p e o f f a u l t p o p u la t i o n c a n n o t b e t e s te d , h o w e v e r , s o t h e

b e s t o n e m a y h o p e fo r i s t o b e a b l e t o fo rm u la t e g e n e ra l ru l e s o r g u id e l in e s fo r u s in g

p a le o s t r e s s a n a l y s i s t e c h n iq u e s (e .g . s t a t i n g t h a t m e t h o d A i s b e t t e r f o r t e s t i n g s m a l l

p o p u la t i o n s o f fa u l t s w h i l e m e th o d B i s b e t t e r fo r l a rg e r o n e s o r m e th o d C c a n n o t b e

u s e d fo r fa u l t p o p u la t i o n s c o n ta in in g c o n ju g a te s e t s o f fa u l t s , e tc . ) .

A n t i c ip a t in g t h e o b l ig a to ry q u e s t io n " I f y o u h a d to d o i t a l l o v e r a g a in , w h a t w o u ld y o u

d o d i f f e r e n t ly ? " , I w o u ld re p ly a s fo l l o w s . I f I h a d to d o i t a l l o v e r a g a in , I w o u ld f i r s t t a k e t h e

t i m e t o w r i te m y o w n c o m p u t e r p r o g r a m s - - i n a l a n g u a g e s u c h a s C a n d f o r a n I B M P C - - t o

p e r fo r m s e v e r a l d i f fe r e n t m e t h o d s o f p a l e o s t r e s s a n a l y s i s . T h e p r o g r a m s w o u l d r e a d t h e d a t a

f ro m a s i n g l e t y p e o f A S C I I f a u l t - s l i p d a t a f i l e a n d w r i te t h e r e s u l t s in a s t a n d a r d i z e d f o r m a t .

O n c e th i s w e re d o n e (n o t a t r i v ia l t a s k ) , t h e t e s t i n g o f t h e se m e th o d s w o u ld b e m u c h e a s i e r th a n

i t i s b y u s in g o th e r p e o p le 's p ro g ra m s . I h a d no w a y o f k n o w in g th i s , h o w e v e r , b e fo re I b e g a n

th i s s tu d y .

1 0 .3 S u g g e s t io n s fo r F u t u r e W o r k

T h e r e i s a g r e a t n e e d f o r m o r e w o r k o n t h e p r o b l e m s o f p a l e o s t r e s s a n a l y s i s . A s m o r e

p e o p le u s e p a le o s t r e s s a n a ly s i s p ro g ra m s , s u c h a s th o s e w r i t t e n b y A n g e l i e r , a g re a te r n e e d

e x i s t s fo r p e o p le t o e v a lu a t e t h e i r e f f e c t i v e n e s s . W h i l e A n g e l i e r m a y b e a w a r e o f th e

l im i t a t i o n s o f h i s m e th o d , th o s e w h o u s e h i s p ro g ra m m a y n o t b e . I r e a l i z e t h a t t h i s t h e s i s s tu d y

i s o n l y a b e g in n in g in t h e s y s t e m a t i c e x a m in a t io n o f p a l e o s t r e s s a n a l y s i s , b u t a t l e a s t i t ' s a

b e g in n in g . S i n c e t h e o b v io u s t im e c o n s t r a in t s fo r m y th e s i s d e fe n s e h a v e p re v e n t e d m e f ro m

in c lu d in g m u c h m o re o f t h e p re l im in a ry d a ta I 'v e g a th e re d , I p la n t o c o n t in u e w i th th e a n a ly se s

o f th e se p a le o s t r e s s p ro g ra m s (p o s s ib ly w i th th e a d d i t io n o f o n e o r t w o m o r e s u c h a s

E t c h e c o p a r 's m e t h o d a n d / o r G e p h a r t a n d F o r s y t h ' s m e t h o d ) w i t h t h e g o a l o f s e n d i n g t h e r e s u l t s

fo r p u b l i c a t i o n b e fo re th e s u m m e r o f 1 9 9 1 .

T h e m a jo r s u g g e s t io n I w o u ld le a v e fo r a n y o n e ( i n c lu d in g m y se l f ) p la n n in g to fu r th e r

e x a m i n e p a l e o s t r e s s a n a l y s i s t e c h n i q u e s , i s to d o w h a t I o u t l in e d i n t h e p r e v i o u s s e c t i o n . W r i te

t h e p r o g r a m s y o u r s e l f a n d s t a n d a r d i z e t h e m . I w o u l d a l s o s u g g e s t q u a n t i fy i n g g r a p h i c a l

m e th o d s o f p a l e o s t r e s s a n a ly s i s s u c h a s L i s le 's (1 9 8 8 ) s o th a t t h e y m a y b e ru n o n a c o m p u te r

a s w e l l . F i n a l ly , t h e c o m p a r i s o n o f t h e d y n a m ic p a l e o s t r e s s a n a ly s i s t e c h n i q u e s t o th e v a r io u s

k i n e m a t i c a n a l y s i s t e c h n i q u e s w h i c h h a v e b e e n p r o p o s e d ( M a r re t t a n d A l l m e n d i n g e r , 1 9 9 0 ) i s

s u re to y i e ld so m e f ru i t fu l i n s ig h t s in to th e m e c h a n ic s o f f a u l t i n g a n d t h e r e l a t i o n s h ip s b e tw e e n

s t r e s s a n d s t r a in .

APPENDIX A

GLOSSARY OF SYMBOLS

T h e f o l l o w i n g i s a l i s t o f m a t h e m a t i c a l s y m b o l s u s e d i n t h i s t h e s i s . T h e s y m b o l s a r e

l i s t e d i n a l p h a b e t i c a l o r d e r w i t h t h e L a t i n a l p h a b e t l i s t e d b e f o r e t h e G r e e k a l p h a b e t . A r b i t r a r y

c o n s ta n t s a r e n o t l i s t e d a n d in c a s e s w h e r e t h e s a m e s y m b o l h a s b e e n u s e d in tw o d i f f e r e n t

c o n te x ts , b o th a re l i s t e d s e p a ra te ly .

0C = Cohesion term of the Coulomb failure criterion

d = Trend of a fault plane's dip direction

NF = Normal force acting upon a plane

SF = Shear force acting upon a plane

TF = Total force acting upon a plane

i = Pitch angle of a fault plane's slip vector

il = Direction cosine where i = 1, 2, or 3

1l = Direction cosine equivalent to l

el = Direction cosine between a fault normal and east

nl = Direction cosine between a fault normal and north

ul = Direction cosine between a fault normal and the vertical axis

2m = Direction cosine equivalent to l

N = Normal force acting upon a plane

N = Normal vector to a fault plane

n = Normal force acting upon a plane

3n = Direction cosine equivalent to l

O = Normal vector to a movement plane

p = A fault plane's dip angle

R = Ratio of the principal stress axes

r = Pitch of a slip vector on a plane

S = Shear force acting upon a plane

S = Slip vector in a fault plane

s = Shear force acting upon a plane

T = Total stress vector acting upon a plane

iT = Total stress vector component i where i = 1, 2, or 3

1t = A positive constant

2t = A positive constant

1,2 2,1" = The shear stress term F or F

2,3 3,2$ = The shear stress term F or F

1,3 3,1( = The shear stress term F or F

* = Tensor aspect ratio of the principal stress axes

* = Dip angle of a plane

, = Meaurement error

0, = Difference between average strike and true strike

S, = Strike measurement error

T, = Trend measurement error

: = Coefficient of friction term of the Coulomb failure criterion

F = The standard deviation of a variable x x

F = The geologic stress tensor

F' = The reduced geologic stress tensor

1F = Most compressive principal stress axis

2F = Intermediate principal stress axis

3F = Least compressive principal stress axis

i,jF = Stress tensor component (i,j) where i and j = 1, 2, or 3

nF = Normal stress acting upon a plane

sF = Shear stress acting upon a plane

xF = Principal stress axis in the X-direction

yF = Principal stress axis in the Y-direction

zF = Principal stress axis in the Z-direction

M = Ratio of the principal stress axes

R = Angular measure modulo 2B

A P P E N D IX B

A R T IF IC IA L F A U L T P O P U L A T I O N D A T A

T h i s i s a l i s t i n g o f a l l t h e a r t i f i c i a l fa u l t p o p u l a t i o n d a t a s h o w n i n s t e r e o g r a p h i c

p ro j e c t io n i n c h a p t e r 8 . T h e c h a n g e i n s l i p v e c to r o r i e n t a t i o n s i s s h o w n fo r e a c h o f th e f i v e

v a l u e s o f M e x a m i n e d ( 0 .0 0 , 0 . 2 5 , 0 . 5 0 , 0 . 7 5 , 1 . 0 0 ) a n d t h e f o r m a t fo r e a c h f a u l t d a t u m i s :

4 6 1 9 2 8 6

w h e re 4 6 ° i s t h e p lu n g e o f th e f a u l t ' s n o rm a l v e c to r , 1 9 2 ° i s t h e t r e n d o f th e f a u l t ' s n o rm a l

v e c t o r , a n d 8 6 ° i s t h e p i t c h o f th e s l i p v e c to r . T h e p i t c h a n g l e i s t h e a n g l e , i n th e p l a n e o f th e

fa u l t , b e tw e e n t h e s t r i k e v e c t o r a n d th e s l i p v e c to r . T h e s t r i k e v e c t o r p o in t s i n th e d i r e c t i o n

d e f in e d b y th e t r e n d o f th e f a u l t p l a n e 's n o r m a l v e c to r + 9 0 ° ( i . e . 2 8 2 ° ) . A p i t c h i s n e g a t i v e i f

t h e a n g l e i s m e a s u r e d f ro m t h e o p p o s i t e s i d e o f t h e s t r ik e v e c t o r ( i . e . t h e t re n d o f t h e f a u l t

p l a n e ' s n o rm a l v e c to r - 9 0 ° ) . A l l o f t h e f a u l t s a r e n o rm a l f a u l t s s o n o a d d i t i o n a l in fo rm a t io n

is n e e d e d re g a rd in g s e n s e -o f - s l i p .

R P -0 1 F A U L T P O P U L A T IO N

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

4 6 1 9 2 9 0 4 6 1 9 2 8 6 4 6 1 9 2 8 2 4 6 1 9 2 7 8 4 6 1 9 2 7 4

2 7 0 1 4 9 0 2 7 0 1 4 8 3 2 7 0 1 4 7 5 2 7 0 1 4 6 8 2 7 0 1 4 6 1

5 2 0 2 5 9 0 5 2 0 2 5 8 3 5 2 0 2 5 7 5 5 2 0 2 5 6 7 5 2 0 2 5 5 9

4 1 3 3 1 9 0 4 1 3 3 1 -8 0 4 1 3 3 1 -7 0 4 1 3 3 1 -5 9 4 1 3 3 1 -5 0

3 9 3 2 6 9 0 3 9 3 2 6 -7 9 3 9 3 2 6 -6 6 3 9 3 2 6 -5 4 3 9 3 2 6 -4 3

2 3 0 2 3 9 0 2 3 0 2 3 7 7 2 3 0 2 3 6 4 2 3 0 2 3 5 2 2 3 0 2 3 4 3

3 5 0 1 3 9 0 3 5 0 1 3 8 4 3 5 0 1 3 7 9 3 5 0 1 3 7 3 3 5 0 1 3 6 8

2 5 3 4 0 9 0 2 5 3 4 0 -7 9 2 5 3 4 0 -6 8 2 5 3 4 0 -5 8 2 5 3 4 0 -4 9

3 1 3 3 9 9 0 3 1 3 3 9 -8 0 3 1 3 3 9 -7 1 3 1 3 3 9 -6 2 3 1 3 3 9 -5 3

3 8 2 1 1 9 0 3 8 2 1 1 7 9 3 8 2 1 1 6 8 3 8 2 1 1 5 6 3 8 2 1 1 4 6

5 5 0 3 6 9 0 5 5 0 3 6 8 1 5 5 0 3 6 7 1 5 5 0 3 6 6 0 5 5 0 3 6 4 8

4 0 0 5 4 9 0 4 0 0 5 4 7 8 4 0 0 5 4 6 1 4 0 0 5 4 4 3 4 0 0 5 4 2 5

4 9 0 1 8 9 0 4 9 0 1 8 8 4 4 9 0 1 8 7 8 4 9 0 1 8 7 3 4 9 0 1 8 6 7

3 4 2 2 1 9 0 3 4 2 2 1 7 6 3 4 2 2 1 6 1 3 4 2 2 1 4 3 3 4 2 2 1 3 3

5 8 0 4 0 9 0 5 8 0 4 0 8 1 5 8 0 4 0 7 0 5 8 0 4 0 5 8 5 8 0 4 0 4 5

3 6 0 2 4 9 0 3 6 0 2 4 8 1 3 6 0 2 4 7 1 3 6 0 2 4 6 2 3 6 0 2 4 5 3

4 1 0 3 0 9 0 4 1 0 3 0 8 0 4 1 0 3 0 6 9 4 1 0 3 0 5 9 4 1 0 3 0 4 9

2 6 3 2 9 9 0 2 6 3 2 9 -7 5 2 6 3 2 9 -6 0 2 6 3 2 9 -4 7 2 6 3 2 9 -3 6

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

3 6 0 0 2 9 0 3 6 0 0 2 8 9 3 6 0 0 2 8 8 3 6 0 0 2 8 7 3 6 0 0 2 8 7

2 4 0 2 2 9 0 2 4 0 2 2 7 8 2 4 0 2 2 6 5 2 4 0 2 2 5 4 2 4 0 2 2 4 5

5 8 1 4 3 9 0 5 8 1 4 3 -8 1 5 8 1 4 3 -7 1 5 8 1 4 3 -6 0 5 8 1 4 3 -4 8

6 1 0 3 0 9 0 6 1 0 3 0 8 2 6 1 0 3 0 7 4 6 1 0 3 0 6 5 6 1 0 3 0 5 7

2 8 2 3 2 9 0 2 8 2 3 2 7 3 2 8 2 3 2 5 3 2 8 2 3 2 3 5 2 8 2 3 2 2 0

5 1 3 5 9 9 0 5 1 3 5 9 9 0 5 1 3 5 9 9 0 5 1 3 5 9 -8 9 5 1 3 5 9 -8 9

5 6 2 2 0 9 0 5 6 2 2 0 8 1 5 6 2 2 0 8 9 5 6 2 2 0 5 7 5 6 2 2 0 4 5

3 5 2 0 5 9 0 3 5 2 0 5 8 0 3 5 2 0 5 6 9 3 5 2 0 5 6 0 3 5 2 0 5 5 1

6 5 1 8 4 9 0 6 5 1 8 4 8 9 6 5 1 8 4 8 8 6 5 1 8 4 8 7 6 5 1 8 4 8 6

4 0 2 2 0 9 0 4 0 2 2 0 7 8 4 0 2 2 0 6 4 4 0 2 2 0 5 0 4 0 2 2 0 3 7

6 0 1 5 2 9 0 6 0 1 5 2 -8 3 6 0 1 5 2 -7 5 6 0 1 5 2 -6 7 6 0 1 5 2 -5 8

3 6 3 4 2 9 0 3 6 3 4 2 -8 3 3 6 3 4 2 -7 5 3 6 3 4 2 -6 8 3 6 3 4 2 -6 1

3 2 3 3 6 9 0 3 2 3 3 6 -8 0 3 2 3 3 6 -6 9 3 2 3 3 6 -5 9 3 2 3 3 6 -5 0

5 0 1 3 6 9 0 5 0 1 3 6 -7 9 5 0 1 3 6 -6 7 5 0 1 3 6 -5 3 5 0 1 3 6 -3 8

3 1 1 6 9 9 0 3 1 1 6 9 -8 5 3 1 1 6 9 -8 0 3 1 1 6 9 -7 4 3 1 1 6 9 -6 9

5 3 1 3 9 9 0 5 3 1 3 9 -8 0 5 3 1 3 9 -6 8 5 3 1 3 9 -5 6 5 3 1 3 9 -4 3

2 8 2 3 5 9 0 2 8 2 3 5 7 3 2 8 2 3 5 5 3 2 8 2 3 5 3 4 2 8 2 3 5 1 8

5 0 1 3 8 9 0 5 0 1 3 8 -8 0 5 0 1 3 8 -6 7 5 0 1 3 8 -4 0 5 0 1 3 8 -4 0

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

2 5 0 2 8 9 0 2 5 0 2 8 7 5 2 5 0 2 8 6 1 2 5 0 2 8 4 9 2 5 0 2 8 3 8

3 8 3 1 2 9 0 3 8 3 1 2 -7 7 3 8 3 1 2 -6 1 3 8 3 1 2 -4 4 3 8 3 1 2 -2 9

3 5 0 3 1 9 0 3 5 0 3 1 7 8 3 5 0 3 1 6 6 3 5 0 3 1 5 4 3 5 0 3 1 4 4

6 4 3 4 9 9 0 6 4 3 4 9 -8 7 6 4 3 4 9 -8 4 6 4 3 4 9 -8 1 6 4 3 4 9 -7 8

2 5 3 1 3 9 0 2 5 3 1 3 -7 1 2 5 3 1 3 -5 1 2 5 3 1 3 -3 4 2 5 3 1 3 -2 2

6 0 1 5 2 9 0 6 0 1 5 2 -8 3 6 0 1 5 2 -7 5 6 0 1 5 2 -6 7 6 0 1 5 2 -5 8

3 1 3 4 4 9 0 3 1 3 4 4 -8 3 3 1 3 4 4 -7 5 3 1 3 4 4 -6 8 3 1 3 4 4 -6 1

5 2 1 8 2 9 0 5 2 1 8 2 8 9 5 2 1 8 2 8 9 5 2 1 8 2 8 8 5 2 1 8 2 8 7

3 0 1 4 8 9 0 3 0 1 4 8 -7 6 3 0 1 4 8 -6 2 3 0 1 4 8 -5 0 3 0 1 4 8 -3 9

4 3 2 1 9 9 0 4 3 2 1 9 7 9 4 3 2 1 9 6 6 4 3 2 1 9 5 3 4 3 2 1 9 4 0

5 3 3 1 9 9 0 5 3 3 1 9 -8 0 5 3 3 1 9 -6 8 5 3 3 1 9 -5 6 5 3 3 1 9 -4 3

2 5 0 4 6 9 0 2 5 0 4 6 7 1 2 5 0 4 6 5 1 2 5 0 4 6 3 5 2 5 0 4 6 2 2

3 5 0 4 9 9 0 3 5 0 4 9 7 6 3 5 0 4 9 5 9 3 5 0 4 9 4 2 3 5 0 4 9 2 7

6 1 1 8 8 9 0 6 1 1 8 8 8 8 6 1 1 8 8 8 5 6 1 1 8 8 8 3 6 1 1 8 8 8 1

4 3 3 0 5 9 0 4 3 3 0 5 -7 8 4 3 3 0 5 -6 3 4 3 3 0 5 -4 4 4 3 3 0 5 -2 6

3 1 3 3 6 9 0 3 1 3 3 6 -7 9 3 1 3 3 6 -6 9 3 1 3 3 6 -5 3 3 1 3 3 6 -4 9

3 1 3 5 3 9 0 3 1 3 5 3 -8 7 3 1 3 5 3 -8 3 3 1 3 5 3 -8 0 3 1 3 5 3 -7 7

2 6 2 9 6 9 0 2 6 2 9 6 -7 4 2 6 2 9 6 -5 3 2 6 2 9 6 -3 0 2 6 2 9 6 -1 2

A C -0 1 F A U L T P O P U L A T IO N

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

3 0 0 0 0 9 0 3 0 0 0 0 9 0 3 0 0 0 0 9 0 3 0 0 0 0 9 0 3 0 0 0 0 9 0

3 0 0 0 5 9 0 3 0 0 0 5 8 8 3 0 0 0 5 8 5 3 0 0 0 5 8 3 3 0 0 0 5 8 0

3 0 3 5 5 9 0 3 0 3 5 5 -8 8 3 0 3 5 5 -8 5 3 0 3 5 5 -8 3 3 0 3 5 5 -8 0

3 0 1 8 0 9 0 3 0 1 8 0 9 0 3 0 1 8 0 9 0 3 0 1 8 0 9 0 3 0 1 8 0 9 0

3 0 1 7 5 9 0 3 0 1 7 5 -8 8 3 0 1 7 5 -8 5 3 0 1 7 5 -8 3 3 0 1 7 5 -8 0

3 0 1 8 5 9 0 3 0 1 8 5 8 8 3 0 1 8 5 8 5 3 0 1 8 5 8 3 3 0 1 8 5 8 0

2 5 0 0 0 9 0 2 5 0 0 0 9 0 2 5 0 0 0 9 0 2 5 0 0 0 9 0 2 5 0 0 0 9 0

2 5 0 0 5 9 0 2 5 0 0 5 8 7 2 5 0 0 5 8 4 2 5 0 0 5 8 1 2 5 0 0 5 7 8

2 5 3 5 5 9 0 2 5 3 5 5 -8 7 2 5 3 5 5 -8 4 2 5 3 5 5 -8 1 2 5 3 5 5 -7 8

2 5 1 8 0 9 0 2 5 1 8 0 9 0 2 5 1 8 0 9 0 2 5 1 8 0 9 0 2 5 1 8 0 9 0

2 5 1 7 5 9 0 2 5 1 7 5 -8 7 2 5 1 7 5 -8 4 2 5 1 7 5 -8 1 2 5 1 7 5 -7 8

2 5 1 8 5 9 0 2 5 1 8 5 8 7 2 5 1 8 5 8 4 2 5 1 8 5 8 1 2 5 1 8 5 7 8

3 5 0 0 0 9 0 3 5 0 0 0 9 0 3 5 0 0 0 9 0 3 5 0 0 0 9 0 3 5 0 0 0 9 0

3 5 0 0 5 9 0 3 5 0 0 5 8 8 3 5 0 0 5 8 6 3 5 0 0 5 8 3 3 5 0 0 5 8 1

3 5 3 5 5 9 0 3 5 3 5 5 -8 8 3 5 3 5 5 -8 6 3 5 3 5 5 -8 3 3 5 3 5 5 -8 1

3 5 1 8 0 9 0 3 5 1 8 0 9 0 3 5 1 8 0 9 0 3 5 1 8 0 9 0 3 5 1 8 0 9 0

3 5 1 7 5 9 0 3 5 1 7 5 -8 8 3 5 1 7 5 -8 6 3 5 1 7 5 -8 3 3 5 1 7 5 -8 1

3 5 1 8 5 9 0 3 5 1 8 5 8 8 3 5 1 8 5 8 6 3 5 1 8 5 8 3 3 5 1 8 5 8 1

A C -0 2 F A U L T P O P U L A T IO N

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

3 0 1 3 5 9 0 3 0 1 3 5 -7 4 3 0 1 3 5 -5 6 3 0 1 3 5 -4 0 3 0 1 3 5 -2 7

3 0 1 4 0 9 0 3 0 1 4 0 -7 5 3 0 1 4 0 -5 8 3 0 1 4 0 -4 3 3 0 1 4 0 -3 1

3 0 1 3 0 9 0 3 0 1 3 0 -7 4 3 0 1 3 0 -5 5 3 0 1 3 0 -3 7 3 0 1 3 0 -2 3

3 0 3 1 5 9 0 3 0 3 1 5 -7 4 3 0 3 1 5 -5 6 3 0 3 1 5 -4 0 3 0 3 1 5 -2 7

3 0 3 1 0 9 0 3 0 3 1 0 -7 4 3 0 3 1 0 -5 5 3 0 3 1 0 -3 7 3 0 3 1 0 -2 3

3 0 3 2 0 9 0 3 0 3 2 0 -7 5 3 0 3 2 0 -5 8 3 0 3 2 0 -4 3 3 0 3 2 0 -3 1

2 5 1 3 5 9 0 2 5 1 3 5 -7 1 2 5 1 3 5 -5 2 2 5 1 3 5 -3 5 2 5 1 3 5 -2 3

2 5 1 4 0 9 0 2 5 1 4 0 -7 2 2 5 1 4 0 -5 4 2 5 1 4 0 -3 8 2 5 1 4 0 -2 7

2 5 1 3 0 9 0 2 5 1 3 0 -7 1 2 5 1 3 0 -5 1 2 5 1 3 0 -3 3 2 5 1 3 0 -2 0

2 5 3 1 5 9 0 2 5 3 1 5 -7 1 2 5 3 1 5 -5 2 2 5 3 1 5 -3 5 2 5 3 1 5 -2 3

2 5 3 1 0 9 0 2 5 3 1 0 -7 1 2 5 3 1 0 -5 1 2 5 3 1 0 -3 3 2 5 3 1 0 -2 0

2 5 3 2 0 9 0 2 5 3 2 0 -7 2 2 5 3 2 0 -5 4 2 5 3 2 0 -3 8 2 5 3 2 0 -2 7

3 5 1 3 5 9 0 3 5 1 3 5 -7 6 3 5 1 3 5 -6 0 3 5 1 3 5 -4 4 3 5 1 3 5 -3 0

3 5 1 4 0 9 0 3 5 1 4 0 -7 7 3 5 1 4 0 -6 2 3 5 1 4 0 -4 7 3 5 1 4 0 -3 4

3 5 1 3 0 9 0 3 5 1 3 0 -7 6 3 5 1 3 0 -5 9 3 5 1 3 0 -4 1 3 5 1 3 0 -2 6

3 5 3 1 5 9 0 3 5 3 1 5 -7 6 3 5 3 1 5 -6 0 3 5 3 1 5 -4 4 3 5 3 1 5 -3 0

3 5 3 1 0 9 0 3 5 3 1 0 -7 6 3 5 3 1 0 -5 9 3 5 3 1 0 -4 1 3 5 3 1 0 -2 6

3 5 3 2 0 9 0 3 5 3 2 0 -7 7 3 5 3 2 0 -6 2 3 5 3 2 0 -4 7 3 5 3 2 0 -3 4

O S -0 1 F A U L T P O P U L A T IO N

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

3 0 0 4 5 9 0 3 0 0 4 5 7 4 3 0 0 4 5 5 6 3 0 0 4 5 4 0 3 0 0 4 5 2 7

3 0 1 3 5 9 0 3 0 1 3 5 -7 4 3 0 1 3 5 -5 6 3 0 1 3 5 -4 0 3 0 1 3 5 -2 7

3 0 2 2 5 9 0 3 0 2 2 5 7 4 3 0 2 2 5 5 6 3 0 2 2 5 4 0 3 0 2 2 5 2 7

3 0 3 1 5 9 0 3 0 3 1 5 -7 4 3 0 3 1 5 -5 6 3 0 3 1 5 -4 0 3 0 3 1 5 -2 7

3 0 0 5 0 9 0 3 0 0 5 0 7 4 3 0 0 5 0 5 5 3 0 0 5 0 3 7 3 0 0 5 0 2 3

3 0 0 4 0 9 0 3 0 0 4 0 7 5 3 0 0 4 0 5 8 3 0 0 4 0 4 3 3 0 0 4 0 3 1

2 5 0 4 5 9 0 2 5 0 4 5 7 1 2 5 0 4 5 5 2 2 5 0 4 5 3 5 2 5 0 4 5 2 3

3 5 0 4 5 9 0 3 5 0 4 5 7 6 3 5 0 4 5 6 0 3 5 0 4 5 4 4 3 5 0 4 5 3 0

3 0 1 3 0 9 0 3 0 1 3 0 -7 4 3 0 1 3 0 -5 5 3 0 1 3 0 -3 7 3 0 1 3 0 -2 3

3 0 1 4 0 9 0 3 0 1 4 0 -7 5 3 0 1 4 0 -5 8 3 0 1 4 0 -4 3 3 0 1 4 0 -3 1

2 5 1 3 5 9 0 2 5 1 3 5 -7 1 2 5 1 3 5 -5 2 2 5 1 3 5 -3 5 2 5 1 3 5 -2 3

3 5 1 3 5 9 0 3 5 1 3 5 -7 6 3 5 1 3 5 -6 0 3 5 1 3 5 -4 4 3 5 1 3 5 -3 0

3 0 2 2 0 9 0 3 0 2 2 0 7 5 3 0 2 2 0 5 8 3 0 2 2 0 4 3 3 0 2 2 0 3 1

3 0 2 3 0 9 0 3 0 2 3 0 7 4 3 0 2 3 0 5 5 3 0 2 3 0 3 7 3 0 2 3 0 2 3

2 5 2 2 5 9 0 2 5 2 2 5 7 1 2 5 2 2 5 5 2 2 5 2 2 5 3 5 2 5 2 2 5 2 3

3 5 2 2 5 9 0 3 5 2 2 5 7 6 3 5 2 2 5 6 0 3 5 2 2 5 4 4 3 5 2 2 5 3 0

3 0 3 1 0 9 0 3 0 3 1 0 -7 4 3 0 3 1 0 -5 5 3 0 3 1 0 -3 7 3 0 3 1 0 -2 3

3 0 3 2 0 9 0 3 0 3 2 0 -7 5 3 0 3 2 0 -5 8 3 0 3 2 0 -4 3 3 0 3 2 0 -3 1

2 5 3 1 5 9 0 2 5 3 1 5 -7 1 2 5 3 1 5 -5 2 2 5 3 1 5 -3 5 2 5 3 1 5 -2 3

3 5 3 1 5 9 0 3 5 3 1 5 -7 6 3 5 3 1 5 -6 0 3 5 3 1 5 -4 4 3 5 3 1 5 -3 0

R S -0 1 F A U L T P O P U L A T IO N

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

3 0 0 0 0 9 0 3 0 0 0 0 9 0 3 0 0 0 0 9 0 3 0 0 0 0 9 0 3 0 0 0 0 9 0

3 0 0 3 0 9 0 3 0 0 3 0 7 7 3 0 0 3 0 6 4 3 0 0 3 0 5 1 3 0 0 3 0 4 1

3 0 0 6 0 9 0 3 0 0 6 0 7 5 3 0 0 6 0 5 5 3 0 0 6 0 3 4 3 0 0 6 0 1 6

3 0 1 2 0 9 0 3 0 1 2 0 -7 5 3 0 1 2 0 -5 5 3 0 1 2 0 -3 4 3 0 1 2 0 -1 6

3 0 1 5 0 9 0 3 0 1 5 0 -7 7 3 0 1 5 0 -6 4 3 0 1 5 0 -5 1 3 0 1 5 0 -4 1

3 0 1 8 0 9 0 3 0 1 8 0 9 0 3 0 1 8 0 9 0 3 0 1 8 0 9 0 3 0 1 8 0 9 0

3 0 2 1 0 9 0 3 0 2 1 0 7 7 3 0 2 1 0 6 4 3 0 2 1 0 5 1 3 0 2 1 0 4 1

3 0 2 4 0 9 0 3 0 2 4 0 7 5 3 0 2 4 0 5 5 3 0 2 4 0 3 4 3 0 2 4 0 1 6

3 0 3 0 0 9 0 3 0 3 0 0 -7 5 3 0 3 0 0 -5 5 3 0 3 0 0 -3 4 3 0 3 0 0 -1 6

3 0 3 3 0 9 0 3 0 3 3 0 -7 7 3 0 3 3 0 -6 4 3 0 3 3 0 -5 1 3 0 3 3 0 -4 1

S O -0 1 F A U L T P O P U L A T IO N

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

3 0 0 0 0 9 0 3 0 0 0 0 9 0 3 0 0 0 0 9 0 3 0 0 0 0 9 0 3 0 0 0 0 9 0

3 5 0 0 0 9 0 3 5 0 0 0 9 0 3 5 0 0 0 9 0 3 5 0 0 0 9 0 3 5 0 0 0 9 0

2 5 0 0 0 9 0 2 5 0 0 0 9 0 2 5 0 0 0 9 0 2 5 0 0 0 9 0 2 5 0 0 0 9 0

3 0 0 0 6 9 0 3 0 0 0 6 8 7 3 0 0 0 6 8 4 3 0 0 0 6 8 1 3 0 0 0 6 7 8

3 5 0 0 6 9 0 3 5 0 0 6 8 7 3 5 0 0 6 8 5 3 5 0 0 6 8 2 3 5 0 0 6 8 0

2 5 0 0 6 9 0 2 5 0 0 6 8 6 2 5 0 0 6 8 3 2 5 0 0 6 7 9 2 5 0 0 6 7 6

3 0 3 5 4 9 0 3 0 3 5 4 -8 7 3 0 3 5 4 -8 4 3 0 3 5 4 -8 1 3 0 3 5 4 -7 8

3 5 3 5 4 9 0 3 5 3 5 4 -8 7 3 5 3 5 4 -8 5 3 5 3 5 4 -8 2 3 5 3 5 4 -8 0

2 5 3 5 4 9 0 2 5 3 5 4 -8 6 2 5 3 5 4 -8 3 2 5 3 5 4 -7 9 2 5 3 5 4 -7 6

3 0 0 0 3 9 0 3 0 0 0 3 8 9 3 0 0 0 3 8 7 3 0 0 0 3 8 6 3 0 0 0 3 8 4

3 5 0 0 3 9 0 3 5 0 0 3 8 9 3 5 0 0 3 8 7 3 5 0 0 3 8 6 3 5 0 0 3 8 5

2 5 0 0 3 9 0 2 5 0 0 3 8 8 2 5 0 0 3 8 6 2 5 0 0 3 8 5 2 5 0 0 3 8 3

3 0 3 5 7 9 0 3 0 3 5 7 -8 9 3 0 3 5 7 -8 7 3 0 3 5 7 -8 6 3 0 3 5 7 -8 4

3 5 3 5 7 9 0 3 5 3 5 7 -8 9 3 5 3 5 7 -8 7 3 5 3 5 7 -8 6 3 5 3 5 7 -8 5

2 5 3 5 7 9 0 2 5 3 5 7 -8 8 2 5 3 5 7 -8 6 2 5 3 5 7 -8 5 2 5 3 5 7 -8 3

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

5 5 0 3 0 9 0 5 5 0 3 0 8 2 5 5 0 3 0 7 3 5 5 0 3 0 6 4 5 5 0 3 0 5 5

5 0 0 3 0 9 0 5 0 0 3 0 8 1 5 0 0 3 0 7 2 5 0 0 3 0 6 2 5 0 0 3 0 5 3

6 0 0 3 0 9 0 6 0 0 3 0 8 2 6 0 0 3 0 7 4 6 0 0 3 0 6 5 6 0 0 3 0 5 6

5 5 0 3 3 9 0 5 5 0 3 3 8 1 5 5 0 3 3 7 2 5 5 0 3 3 6 2 5 5 0 3 3 5 2

5 0 0 3 3 9 0 5 0 0 3 3 8 1 5 0 0 3 3 7 1 5 0 0 3 3 6 0 5 0 0 3 3 5 0

6 0 0 3 3 9 0 6 0 0 3 3 8 2 6 0 0 3 3 7 3 6 0 0 3 3 6 3 6 0 0 3 3 5 3

5 5 0 3 6 9 0 5 5 0 3 6 8 2 5 5 0 3 6 7 1 5 5 0 3 6 6 0 5 5 0 3 6 4 8

5 0 0 3 6 9 0 5 0 0 3 6 8 0 5 0 0 3 6 6 9 5 0 0 3 6 5 8 5 0 0 3 6 4 7

6 0 0 3 6 9 0 6 0 0 3 6 8 1 6 0 0 3 6 7 2 6 0 0 3 6 6 1 6 0 0 3 6 5 0

5 5 0 2 7 9 0 5 5 0 2 7 8 3 5 5 0 2 7 7 5 5 5 0 2 7 6 6 5 5 0 2 7 5 8

5 0 0 2 7 9 0 5 0 0 2 7 8 2 5 0 0 2 7 7 4 5 0 0 2 7 6 5 5 0 0 2 7 5 6

6 0 0 2 7 9 0 6 0 0 2 7 8 3 6 0 0 2 7 7 5 6 0 0 2 7 6 7 6 0 0 2 7 6 0

5 5 0 2 4 9 0 5 5 0 2 4 8 3 5 5 0 2 4 7 6 5 5 0 2 4 6 9 5 5 0 2 4 6 1

5 0 0 2 4 9 0 5 0 0 2 4 8 3 5 0 0 2 4 7 5 5 0 0 2 4 6 7 5 0 0 2 4 6 0

6 0 0 2 4 9 0 6 0 0 2 4 8 4 6 0 0 2 4 7 7 6 0 0 2 4 7 0 6 0 0 2 4 6 3

M = 0 .0 0 M = 0 .2 5 M = 0 .5 0 M = 0 .7 5 M = 1 .0 0

3 0 0 4 5 9 0 3 0 0 4 5 7 4 3 0 0 4 5 5 6 3 0 0 4 5 4 0 3 0 0 4 5 2 7

2 5 0 4 5 9 0 2 5 0 4 5 7 1 2 5 0 4 5 5 2 2 5 0 4 5 3 5 2 5 0 4 5 2 3

3 5 0 4 5 9 0 3 5 0 4 5 7 6 3 5 0 4 5 6 0 3 5 0 4 5 4 4 3 5 0 4 5 3 0

3 0 0 4 8 9 0 3 0 0 4 8 7 4 3 0 0 4 8 5 6 3 0 0 4 8 3 8 3 0 0 4 8 2 4

2 5 0 4 8 9 0 2 5 0 4 8 7 1 2 5 0 4 8 5 1 2 5 0 4 8 3 4 2 5 0 4 8 2 1

3 5 0 4 8 9 0 3 5 0 4 8 7 6 3 5 0 4 8 5 9 3 5 0 4 8 4 2 3 5 0 4 8 2 7

3 0 0 5 1 9 0 3 0 0 5 1 7 4 3 0 0 5 1 5 5 3 0 0 5 1 3 7 3 0 0 5 1 2 2

2 5 0 5 1 9 0 2 5 0 5 1 7 1 2 5 0 5 1 5 0 2 5 0 5 1 3 2 2 5 0 5 1 1 9

3 5 0 5 1 9 0 3 5 0 5 1 7 6 3 5 0 5 1 5 9 3 5 0 5 1 4 1 3 5 0 5 1 2 5

3 0 0 4 2 9 0 3 0 0 4 2 7 4 3 0 0 4 2 5 7 3 0 0 4 2 4 2 3 0 0 4 2 2 9

2 5 0 4 2 9 0 2 5 0 4 2 7 2 2 5 0 4 2 6 3 2 5 0 4 2 3 7 2 5 0 4 2 2 5

3 5 0 4 2 9 0 3 5 0 4 2 7 6 3 5 0 4 2 6 1 3 5 0 4 2 4 6 3 5 0 4 2 3 3

3 0 0 3 9 9 0 3 0 0 3 9 7 5 3 0 0 3 9 5 9 3 0 0 3 9 4 4 3 0 0 3 9 3 2

2 5 0 3 9 9 0 2 5 0 3 9 7 3 2 5 0 3 9 5 4 2 5 0 3 9 3 9 2 5 0 3 9 2 8

3 5 0 3 9 9 0 3 5 0 3 9 7 6 3 5 0 3 9 6 2 3 5 0 3 9 4 8 3 5 0 3 9 3 5

A P P E N D IX C

S L IP V E C T O R C A L C U L A T IO N P R O G R A M

C o m p le t e l i s t i n g o f th e s l i p v e c t o r c a l c u l a t i o n p ro g ra m d i s c u s s e d in c h a p t e r f i v e . T h e

p ro g ra m c o n s i s t s o f th re e f i l e s - - t h e m a in p ro g ra m S L IP .P A S , t h e i n c lu d e f i l e D X F . IN C w h ic h

c o n ta in s A u to C A D D X F f i l e c r e a t i o n p ro c e d u re s , a n d th e h e lp f i l e S L IP .T X T . T h e p ro g ra m

is w r i t t e n in T u rb o P a s c a l v e rs io n 3 .0 1 fo r a n IB M P C o r c o m p a t i b le c o m p u te r .

program Slip;

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * }

{ This is a program to calculate the slip vector and stress ratios }

{ on planes of random orientations in a stress field given the mean }

{ and deviatoric stresses. Read the help file SLIP.TXT to learn more }

{ about this program. }

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * }

{ SLIP.PAS - Version 2.0 }

{ For educational and research purposes only }

{ All commercial rights reserved }

{ Steven H. Schimmrich }

{ Department of Geological Sciences }

{ State University of New York at Albany }

{ Albany, New York 12222 }

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * }

{ Initializations }

label 1;

HELPFILE = 'SLIP.TXT';

PAGES = 5;

PI = 3.1415927;

RegisterList =

record

AX,BX,CX,DX,BP,SI,DI,DS,ES,Flags : integer;

linarray2d = array[1..3,1..51] of real;

chararray3 = array[1..3] of string[5];

linarray51 = array[1..51] of real;

linarray3 = array[1..3] of real;

string12 = string[12];

string78 = string[78];

Sigma1, Sigma3, Mu, Cohesion,

Plunge, Trend, CutOff : real;

DMin, DMax : char;

Choice, Intervals : integer;

Cosines : linarray3;

Sigma : chararray3;

Stresses, SlipVector : linarray2d;

Ratios, SlipAngle : linarray51;

QuitIt, IsThereData : boolean;

FileName : string12;

{ General program functions }

function Exists(FileName : string12): boolean;

{ Checks to see if a file exists on the disk }

Name : file;

Assign(Name,FileName);

Reset(Name);

Exists := (IOResult = 0);

function Zero(Value : real): boolean;

{ Checks to see if a value is essentially 0.0 }

if (abs(Value) < 0.00001)

then Zero := true

else Zero := false;

function DegToRad(DegreeMeasure : real): real;

{ Converts an angle in degrees to one in radians }

DegToRad := ((DegreeMeasure * PI) / 180.0);

function RadToDeg(RadianMeasure : real): real;

{ Converts an angle in radians to one in degrees }

RadToDeg := ((RadianMeasure * 180.0) / PI);

function Tan(Angle : real): real;

{ Returns the tangent of an angle }

Tan := sin(Angle) / cos(Angle);

function ArcCos(AValue : real): real;

{ Returns the arccosine of a value }

X, Y : real;

if (AValue = 0.0)

then ArcCos := (PI / 2.0)

else if (AValue = 1.0)

then ArcCos := 0.0

else if (AValue = -1.0)

then ArcCos := PI

X := (AValue / sqrt(1.0 - sqr(AValue)));

Y := arctan(abs(1.0 / X));

if (X > 0.0)

then ArcCos := Y

else ArcCos := (PI - Y);

function ArcSin(AValue : real): real;

{ Returns the arcsine of a value }

X, Y : real;

if (AValue = 0.0)

then ArcSin := 0.0

if (AValue = 1.0)

then ArcSin := (PI / 2.0)

if (AValue = -1.0)

then ArcSin := (-PI / 2.0)

if (X = 0.0)

then ArcSin := 0.0

Y := arctan(abs(X));

if (X > 0.0)

then ArcSin := Y

else ArcSin := -Y;

{ General program procedures }

procedure Alarm;

{ Sounds an alarm }

Count : integer;

for Count := 1 to 4 do

sound(880);

delay(50);

sound(0);

delay(50);

nosound;

procedure Beep;

{ Sounds a beep }

sound(880);

delay(50);

nosound;

procedure HoldScreen(var QuitIt : boolean);

{ Holds the screen until any key is pressed }

Key : char;

write(' Press any ');

textcolor(12);

write('key');

textcolor(14);

write(' to continue (');

textcolor(12);

write('Q');

textcolor(14);

write(' to quit)...');

read(kbd,Key);

if (upcase(Key) = 'Q')

then QuitIt := true

else QuitIt := false;

procedure EndGraphics;

{ Ends a graphics display }

repeat until (keypressed);

textmode;

textcolor(11);

clrscr;

procedure Cursor(On : boolean);

{ Turns cursor on and off }

Register : RegisterList;

if (On)

if (mem[0:$449] = 7)

Register.CX := $0C0D

Register.CX := $0607

Register.CX := $2000;

Register.AX := $0100;

intr($10,Register);

procedure DrawBox(ULX, ULY, LRX, LRY : integer);

{ Draws a box around text in text mode }

X, Y, XDistance : integer;

gotoxy(ULX,ULY);

write(#201);

XDistance := LRX - ULX - 1;

for X := 1 to XDistance do

write(#205);

write(#187);

for Y := (ULY + 1) to (LRY - 1) do

gotoxy(LRX,Y);

write(#186);

gotoxy(ULX,Y);

write(#186);

gotoxy(ULX,LRY);

write(#200);

write(#205);

write(#188);

{ Error display procedure }

procedure ShowError(ErrorType : integer);

{ Displays appropriate error messages }

clrscr;

Alarm;

gotoxy(28,2);

textcolor(12);

writeln('Slip Vector Plotting Program');

gotoxy(37,10);

textcolor(28);

writeln('* ERROR *');

textcolor(11);

case (ErrorType) of

1 : begin

gotoxy(21,14);

write('File ',HELPFILE,' does not exist on the disk');

2 : begin

gotoxy(22,14);

writeln('F1 and F3 must be between -100 and +100');

3 : begin

gotoxy(30,14);

writeln('F3 must be less than F1');

4 : begin

gotoxy(26,14);

writeln(': must be between 0.0 and 100.0');

5 : begin

gotoxy(26,14);

writeln('C must be between 0.0 and 100.0');

6 : begin

gotoxy(20,14);

writeln('The plunge must be between 0 and 90 degrees');

7 : begin

gotoxy(20,14);

writeln('The trend must be between 0 and 360 degrees');

8 : begin

gotoxy(20,14);

writeln('Can only examine between 2 and 50 intervals');

9 : begin

gotoxy(19,14);

writeln('Do not specify an extension for the filename');

10 : begin

gotoxy(23,14);

writeln('That file already exists on this disk');

11 : begin

gotoxy(29,14);

writeln('Data must be entered first');

12 : begin

gotoxy(18,14);

writeln('The number entered must be a positive real value');

13 : begin

gotoxy(21,14);

writeln('F1 and F3 cannot have the same orientation');

gotoxy(30,15);

write('Recheck and try again...');

delay(5000);

{ Informational page display procedures }

procedure IntroPage;

{ Displays an introduction page }

textmode;

clrscr;

cursor(false);

textcolor(12);

DrawBox(20,9,61,18);

textcolor(11);

gotoxy(27,10);

gotoxy(32,12);

writeln('Program written by');

gotoxy(31,14);

writeln('Steven H. Schimmrich');

gotoxy(24,15);

writeln('Department of Geological Sciences');

gotoxy(22,16);

writeln('State University of New York at Albany');

gotoxy(30,17);

writeln('Albany, New York 12222');

textcolor(11);

delay(5000);

cursor(true);

procedure Working;

{ Displays a message that the program is working }

clrscr;

Cursor(false);

textcolor(12);

DrawBox(26,10,57,16);

textcolor(11);

gotoxy(28,11);

gotoxy(36,13);

writeln('Don''t Panic!');

gotoxy(33,15);

writeln('I''m working on it');

delay(1000);

procedure W riting(FileName : string12);

{ Displays a message that the program is writing to a file }

Len1, Len2 : integer;

clrscr;

Len1 := length(FileName);

Len2 := trunc((12 - Len1) / 2);

cursor(false);

textcolor(12);

DrawBox((21 + Len2),9,(50 + Len1 + Len2),15);

textcolor(11);

gotoxy(28,10);

gotoxy(36,12);

gotoxy((23 + Len2),14);

write('Writing data to disk file ');

textcolor(12);

write(FileName);

textcolor(11);

delay(1000);

procedure CreatePage(FileName : string12);

{ Displays a page that DXF is being created for AutoCAD }

Len1, Len2 : integer;

clrscr;

Len1 := length(FileName);

Len2 := trunc((12 - Len1) / 2);

cursor(false);

textcolor(12);

DrawBox((18 + Len2),9,(52 + Len1 + Len2),15);

textcolor(11);

gotoxy(28,10);

gotoxy(36,12);

gotoxy((20 + Len2),14);

write('Writing data to AutoCAD DXFile ');

textcolor(12);

write(FileName);

textcolor(11);

delay(1000);

procedure ExitPage;

{ Displays an exit page }

textmode;

clrscr;

cursor(false);

textcolor(12);

DrawBox(26,10,57,16);

textcolor(11);

gotoxy(28,11);

gotoxy(35,13);

writeln('POET Software');

gotoxy(33,14);

gotoxy(32,15);

delay(5000);

cursor(true);

textmode;

clrscr;

{ Main menu procedures }

procedure MainMenuChoice(var Choice : integer);

{ Returns the number of the menu item selected }

Key : char;

repeat

read(kbd,Key);

Choice := ord(Key) - 48;

if (not(Choice in [1..7]))

then Beep;

until (Choice in [1..7]);

procedure MainMenu(var Choice : integer);

{ Displays main operations menu }

Count : integer;

clrscr;

textcolor(12);

DrawBox(25,2,56,4);

textcolor(11);

gotoxy(27,3);

gotoxy(2,9);

writeln('What do you wish to do ?');

writeln;

textcolor(12);

write(' ',Count);

textcolor(11);

case (Count) of

1 : writeln(' - Learn more about the program');

2 : writeln(' - Perform the slip vector calculations');

3 : writeln(' - Display a graph of the results');

4 : writeln(' - Display a stereonet of the results');

5 : writeln(' - Display a listing of the results');

6 : writeln(' - Print the results');

7 : writeln(' - Exit the program');

writeln;

write(' Press the ');

textcolor(12);

write('number');

textcolor(11);

write(' of you choice...');

MainMenuChoice(Choice);

{ Main menu option # 1 - Help pages display }

procedure HelpPages;

{ Sequentially displays help file }

label 1;

Line : string78;

Page, Row : integer;

Key : char;

DataFile : text[1024];

assign(DataFile,HELPFILE);

reset(DataFile);

for Row := 1 to 5 do

readln(DataFile,Line);

for Page := 1 to PAGES do

clrscr;

for Row := 1 to 23 do

readln(DataFile,Line);

if (copy(Line,1,1) = '*')

textcolor(12);

delete(Line,1,1);

writeln(Line);

textcolor(11);

write(' Press any ');

textcolor(12);

write('key');

textcolor(11);

write(' to continue (');

textcolor(12);

write('Q');

textcolor(11);

write(' to quit)...');

read(kbd,Key);

if (upcase(Key) = 'Q')

then goto 1;

1:close(DataFile);

{ Main menu option # 2 - Perform calculations }

procedure AskData(var Sigma : chararray3; var Sigma1, Sigma3, Mu, Cohesion, Plunge,

Trend : real; var DMin, DMax : char; var Intervals : integer);

{ Asks for user supplied data }

label 1;

1:{Continue}

clrscr;

textcolor(12);

DrawBox(25,1,56,3);

gotoxy(27,2);

textcolor(11);

gotoxy(2,5);

writeln('What is the orientation of the maximum compressive');

gotoxy(2,6);

write('principal stress axis F1 (');

textcolor(12);

write('N');

textcolor(11);

write('orth, ');

textcolor(12);

write('E');

textcolor(11);

write('ast, or ');

textcolor(12);

write('U');

textcolor(11);

write('p) ? ');

repeat

read(kbd,DMax);

if (not(upcase(DMax) in ['N','E','U']))

then Beep;

until (upcase(DMax) in ['N','E','U']);

gotoxy(50,6);

textcolor(12);

writeln(upcase(DMax));

textcolor(11);

gotoxy(2,8);

writeln('What is the orientation of the minimum compressive');

gotoxy(2,9);

write('principal stress axis F3 (');

textcolor(12);

write('N');

textcolor(11);

write('orth, ');

textcolor(12);

write('E');

textcolor(11);

write('ast, or ');

textcolor(12);

write('U');

textcolor(11);

write('p) ? ');

repeat

read(kbd,DMin);

if (not(upcase(DMin) in ['N','E','U']))

then Beep;

until (upcase(DMin) in ['N','E','U']);

gotoxy(50,9);

textcolor(12);

writeln(upcase(DMin));

textcolor(11);

if (upcase(Dmax) = upcase(DMin))

ShowError(13);

goto 1;

case (upcase(DMax)) of

'N' : Sigma[1] := 'North';

'U' : Sigma[1] := ' Up';

'E' : Sigma[1] := ' East';

case (upcase(DMin)) of

'N' : Sigma[3] := 'North';

'U' : Sigma[3] := ' Up';

'E' : Sigma[3] := ' East';

if ((not(upcase(DMin) in ['N'])) and (not(upcase(DMax) in ['N'])))

then Sigma[2] := 'North';

if ((not(upcase(DMin) in ['U'])) and (not(upcase(DMax) in ['U'])))

then Sigma[2] := ' Up';

if ((not(upcase(DMin) in ['E'])) and (not(upcase(DMax) in ['E'])))

then Sigma[2] := ' East';

gotoxy(2,11);

write('Enter the value for F1 : ');

Sigma1 := -999.9;

textcolor(12);

readln(Sigma1);

textcolor(11);

if ((Sigma1 < -100.0) or (Sigma1 > 100.0))

ShowError(2);

goto 1;

gotoxy(2,12);

write('Enter the value for F3 : ');

Sigma3 := -999.9;

textcolor(12);

readln(Sigma3);

textcolor(11);

if ((Sigma3 < -100.0) or (Sigma3 > 100.0))

ShowError(2);

goto 1;

if (Sigma3 > Sigma1)

ShowError(3);

goto 1;

gotoxy(2,14);

write('Enter the coefficient of friction (:) : ');

textcolor(12);

Mu := -999.9;

readln(Mu);

textcolor(11);

if ((Mu < 0.0) or (Mu > 100.0))

ShowError(4);

goto 1;

gotoxy(2,15);

write('Enter the cohesion (C) : ');

textcolor(12);

Cohesion := -999.9;

readln(Cohesion);

textcolor(11);

if ((Cohesion < 0.0) or (Cohesion > 100.0))

ShowError(5);

goto 1;

gotoxy(2,17);

writeln('Now enter the plunge and trend of the normal');

writeln(' vector to the fault plane you wish to examine');

gotoxy(2,20);

write('Enter the plunge : ');

Plunge := -999.9;

textcolor(12);

readln(Plunge);

textcolor(11);

if ((Plunge < 0.0) or (Plunge >= 90.0))

ShowError(6);

goto 1;

gotoxy(2,21);

write('Enter the trend : ');

Trend := -999.9;

textcolor(12);

readln(Trend);

textcolor(11);

if ((Trend < 0.0) or (Trend >= 360.0))

ShowError(7);

goto 1;

if (Trend = 360.0)

then Trend := Trend - 360.0;

gotoxy(2,23);

write('How many values of F2 between F1 and F3 do you wish to examine ? ');

Intervals := -9;

textcolor(12);

readln(Intervals);

textcolor(11);

if ((Intervals < 2) or (Intervals > 50))

ShowError(8);

goto 1;

procedure DirCosines(Plunge, Trend : real; DMin, DMax : char; var Cosines : linarray3);

{ Returns the direction cosines of the normal vector }

Plunge := DegToRad(Plunge);

Trend := DegToRad(Trend);

if ((upcase(DMax) = 'N') and (upcase(DMin) = 'E'))

Cosines[1] := (cos(Plunge) * cos(Trend));

Cosines[2] := sin(Plunge);

Cosines[3] := (cos(Plunge) * sin(Trend));

if ((upcase(DMax) = 'N') and (upcase(DMin) = 'U'))

if ((upcase(DMax) = 'E') and (upcase(DMin) = 'N'))

if ((upcase(DMax) = 'E') and (upcase(DMin) = 'U'))

if ((upcase(DMax) = 'U') and (upcase(DMin) = 'N'))

if ((upcase(DMax) = 'U') and (upcase(DMin) = 'E'))

procedure Cauchy(Intervals : integer; Sigma1, Sigma3 : real; Cosines : linarray3;

var Stresses : linarray2d);

{ Uses Cauchy's Formula to calculate the total stress components }

Count, Value : integer;

Sigma2, Step, Temp, Deviator : real;

Deviator := (Sigma1 - Sigma3);

Step := (Deviator / Intervals);

Sigma2 := Sigma3;

for Count := 1 to (Intervals + 1) do

Stresses[1,Count] := Sigma1 * Cosines[1];

Sigma2 := Sigma2 + Step;

procedure CalculateStresses(Intervals : integer; Mu, Cohesion, Trend, Plunge : real;

Cosines : linarray3; Stresses : linarray2d;

var SlipVector : linarray2d; DMin, DMax : char;

var SlipAngle, Ratios : linarray51);

{ Calculates slip vector and shear to normal stress ratio for each phi }

Count : integer;

DotProduct, StressVectorNormalized, Angle,

ShearStress, NormalStress, Phi, Sigma1,

Sigma2, Sigma3, Beta, Strike : real;

StressVectorNormalized := (sqrt(sqr(Stresses[1,Count]) + sqr(Stresses[2,Count]) +

sqr(Stresses[3,Count])));

DotProduct := ((Cosines[1] * Stresses[1,Count]) + (Cosines[2] * Stresses[2,Count]) +

(Cosines[3] * Stresses[3,Count]));

Angle := (ArcCos(DotProduct / StressVectorNormalized));

ShearStress := (abs(StressVectorNormalized * sin(Angle)));

NormalStress := (abs(StressVectorNormalized * cos(Angle)));

if (((Zero(NormalStress)) or (Zero(Mu))) and (Zero(Cohesion)))

then Ratios[Count] := 0.0

else Ratios[Count] := (ShearStress / ((NormalStress * Mu) + Cohesion));

if (Cosines[1] <> 0.0)

then Sigma1 := Stresses[1,Count] / Cosines[1]

else Sigma1 := 0.0;

else Sigma2 := 0.0;

else Sigma3 := 0.0;

if ((upcase(DMax) = 'N') and (upcase(DMin) = 'E'))

Phi := ((Sigma2 - Sigma3) / (Sigma1 - Sigma3));

if (Zero(Cosines[2]))

then SlipAngle[Count] := 0.0;

then SlipAngle[Count] := (PI / 2.0);

if ((not(Zero(Cosines[1]))) and (not(Zero(Cosines[2]))) and

(not(Zero(Cosines[3]))))

then SlipAngle[Count] := arctan(((sqr(Cosines[1]) * Cosines[2]) - (Phi * Cosines[2]) +

(Phi * sqr(Cosines[2] * Cosines[2])) / (Cosines[3] * Cosines[1]));

if ((upcase(DMax) = 'N') and (upcase(DMin) = 'U'))

if (Sigma1 = Sigma2)

then SlipAngle[Count] := (PI / 2.0)

then SlipAngle[Count] := arctan(((sqr(Cosines[1]) * Cosines[3]) - (Phi * Cosines[3])

+ (Phi * sqr(Cosines[3]) * Cosines[3])) /

(Cosines[2] * Cosines[1]));

if ((upcase(DMax) = 'U') and (upcase(DMin) = 'N'))

if ((upcase(DMax) = 'U') and (upcase(DMin) = 'E'))

if ((upcase(DMax) = 'E') and (upcase(DMin) = 'U'))

if ((upcase(DMax) = 'E') and (upcase(DMin) = 'N'))

if ((not(Zero(Cosines[1]))) and (not(Zero(Cosines[2]))) and (not(Zero(Cosines[3]))))

then SlipAngle[Count] := arctan(((sqr(Cosines[3]) * Cosines[2]) - (Phi * Cosines[2]) +

(Phi * sqr(Cosines[2]) * Cosines[2])) /

SlipAngle[Count] := RadToDeg(SlipAngle[Count]);

if (not(Zero(SlipAngle[Count])))

then Beta := RadToDeg(arctan(cos(DegToRad(90 - Plunge)) *

Tan(DegToRad(SlipAngle[Count]))))

else Beta := 0.0;

Strike := Trend + 90.0;

if (Strike > 360.0)

then Strike := Strike - 180.0;

SlipVector[2,Count] := Strike + Beta;

if (not(Zero(abs(Beta) - 90.0)))

then SlipVector[1,Count] := RadToDeg(ArcCos(cos(DegToRad(SlipAngle[Count])) /

cos(DegToRad(Beta))))

else SlipVector[1,Count] := 90.0 - Plunge;

if ((SlipAngle[Count] < 0.0) or (SlipAngle[Count] = 90.0))

then SlipVector[2,Count] := SlipVector[2,Count] + 180.0;

if (Trend > 270)

then SlipVector[2,Count] := SlipVector[2,Count] + 180.0;

if (SlipVector[2,Count] > 360.0)

then SlipVector[2,Count] := SlipVector[2,Count] - 360.0;

{ Main menu option # 3 - Examine a graph of the results }

procedure Circle(XVal, YVal : integer; Radius : real);

{ Draws a circle of a given radius about a given center point }

Angle, X, Y : integer;

Radian : real;

for Angle := 0 to 180 do

Radian := (((Angle * 2.0) * PI) / 180.0);

Y := round(YVal - (Radius * cos(Radian)));

X := round(XVal + (Radius * (sin(Radian) / 0.416)));

plot(X,Y,1);

procedure AskCutOff(var CutOff : real);

{ Asks cutoff value for the shear to normal stress ratios }

repeat

clrscr;

textcolor(12);

DrawBox(25,1,56,3);

textcolor(11);

gotoxy(27,2);

gotoxy(2,8);

write('Enter minimum shear to normal stress ratio (default is 0.0) : ');

textcolor(12);

CutOff := 0.0;

readln(CutOff);

textcolor(11);

if (CutOff < 0.0)

then ShowError(12);

until (CutOff >= 0.0);

procedure DrawGraphAxes(CutOff, Trend, Plunge : real; Sigma : chararray3);

{ Draws the axes of the graph }

IntPlunge, IntTrend : integer;

hires;

palette(1);

hirescolor(4);

gotoxy(2,1);

gotoxy(2,2);

IntPlunge := round(Plunge);

IntTrend := round(Trend);

writeln('Fault plane normal at ',IntPlunge:2,' / ',IntTrend:3);

gotoxy(2,3);

writeln('Cutoff value = ',CutOff:4:2);

gotoxy(40,1);

writeln('Fault Planes');

Circle(318,11,2.0);

gotoxy(42,2);

writeln(' = Slipped');

Circle(318,20,1.0);

gotoxy(42,3);

writeln(' = Locked');

gotoxy(62,1);

writeln('Sigma 1 = ',Sigma[1]);

gotoxy(62,2);

gotoxy(62,3);

draw(71,115,590,115,1);

gotoxy(2,24);

writeln('Phi');

gotoxy(9,24);

writeln('0.0');

draw(178,113,178,117,1);

gotoxy(22,24);

writeln('0.2');

draw(281,113,281,117,1);

gotoxy(35,24);

writeln('0.4');

draw(384,113,384,117,1);

gotoxy(48,24);

writeln('0.6');

draw(487,113,487,117,1);

gotoxy(61,24);

writeln('0.8');

draw(590,113,590,117,1);

gotoxy(74,24);

writeln('1.0');

draw(75,180,75,50,1);

gotoxy(2,5);

writeln('Pitch');

draw(79,180,71,180,1);

gotoxy(6,23);

writeln('-90');

draw(79,147,71,147,1);

gotoxy(6,19);

writeln('-45');

draw(79,115,71,115,1);

gotoxy(8,15);

writeln('0');

draw(79,83,71,83,1);

gotoxy(7,11);

writeln('45');

draw(79,50,71,50,1);

gotoxy(7,7);

writeln('90');

procedure PlotPoints(Intervals : integer; CutOff, Sigma1, Sigma3 : real; Ratios,

SlipAngle : linarray51);

{ Plots the phi versus pitch points }

Count, XVal, YVal : integer;

Deviator, Phi, Step, Sigma2 : real;

Sigma2 := Sigma3;

XVal := round(75.0 + (Phi * 515.0));

YVal := round(115.0 - ((SlipAngle[Count] * 130.0) / 180.0));

if (Ratios[Count] >= CutOff)

then Circle(XVal,YVal,2.0)

else Circle(XVal,YVal,1.0);

if (abs(abs(SlipAngle[Count]) - 90.0) < 0.0001)

YVal := round(115.0 - ((abs(SlipAngle[Count]) * 130.0) / 180.0));

YVal := round(115.0 - ((-abs(SlipAngle[Count]) * 130.0) / 180.0));

{ Main menu option # 4 - Display a stereonet of the results }

procedure DrawStereonet(CutOff, Plunge, Trend : real; Sigma : chararray3);

{ Draws stereonet on the screen }

Angle, X, Y, IntPlunge, IntTrend : integer;

Radian : real;

hires;

palette(1);

hirescolor(4);

for Angle := 0 to 3600 do

Radian := DegToRad(Angle / 10.0);

X := round(315.0 + (90.0 * (sin(Radian) / 0.416)));

Y := round(102.0 - (90.0 * cos(Radian)));

plot(X,Y,1);

draw(315,100,315,104,1);

draw(313,102,317,102,1);

draw(315,14,315,10,1);

gotoxy(40,1);

writeln('N');

gotoxy(3,2);

writeln('Slip Vector');

gotoxy(3,3);

writeln('Plotting Program');

gotoxy(3,4);

writeln('Fault Plane Normal');

gotoxy(3,5);

writeln('at ',IntPlunge:2,' / ',IntTrend:3);

gotoxy(63,2);

writeln('Lower-Hemisphere');

gotoxy(66,3);

writeln('Stereographic');

gotoxy(69,4);

writeln('Projection');

gotoxy(63,22);

gotoxy(63,23);

gotoxy(63,24);

gotoxy(5,22);

writeln('= Slip');

gotoxy(5,23);

writeln('= Locked');

gotoxy(3,24);

writeln('Cutoff Value = ',CutOff:4:2);

circle(18,170,2.0);

circle(18,179,1.0);

procedure DrawFaultPlane(Plunge, Trend, Radius : real);

{ Draws a great circle on a Wulff net }

Step, XVal, YVal : integer;

Strike, Dip, ApparentDip, Distance, Beta : real;

if (Plunge = 0.0)

then Plunge := Plunge + 0.0001;

Dip := DegToRad(90.0 - Plunge);

if ((Trend >= 90.0) and (Trend < 270.0))

then Strike := Trend - 90.0

else Strike := Trend + 90.0;

if (Strike >= 360.0)

for Step := 0 to 1800 do

Beta := DegToRad(Step / 10.0);

ApparentDip := arctan(Tan(Dip) * sin(Beta));

Distance := Radius * Tan((PI / 4.0) - (ApparentDip / 2.0));

if (Distance < 1.0)

Distance := RadToDeg(Beta);

if (Distance > 90.0)

then Distance := 90.0 - Distance;

Beta := DegToRad(Strike);

Beta := (((5.0 * PI) / 2.0) - Beta);

if (Beta >= (2.0 * PI))

then Beta := (Beta - (2.0 * PI));

if (Beta < 0.0)

then Beta := (Beta + (2.0 * PI));

then Beta := DegToRad(Strike) - Beta

else Beta := DegToRad(Strike) + Beta;

if (Beta >= (2.0 * PI))

if (Beta < 0.0)

Beta := (((5.0 * PI) / 2.0) - Beta);

if (Beta >= (2.0 * PI))

if (Beta < 0.0)

XVal := round(315.0 + ((Distance * cos(Beta)) / 0.416));

YVal := round(102.0 - (Distance * sin(Beta)));

plot(XVal,YVal,1);

procedure DataPlot(Intervals : integer; CutOff, Trend : real; Cosines : linarray3;

SlipVector : linarray2d; Ratios : linarray51);

{ Plots slip vectors on the fault plane }

Count, XVal, YVal : integer;

Distance, Theta, ZeroTrend : real;

Distance := (90.0 * Tan((PI / 4.0) - (DegToRad(SlipVector[1,Count]) / 2.0)));

if (SlipVector[1,Count] = 0.0)

ZeroTrend := (SlipVector[2,Count] + 180.0);

if (ZeroTrend >= 360.0)

then ZeroTrend := ZeroTrend - 360.0;

Theta := (((5.0 * PI) / 2.0) - DegToRad(ZeroTrend));

XVal := round(315.0 + ((Distance * cos(Theta)) / 0.416));

YVal := round(102.0 - (Distance * sin(Theta)));

Theta := (((5.0 * PI) / 2.0) - DegToRad(SlipVector[2,Count]));

XVal := round(315.0 + ((Distance * cos(Theta)) / 0.416));

YVal := round(102.0 - (Distance * sin(Theta)));

if ((Count = 1) and (not(Zero(Cosines[1]))) and (not(Zero(Cosines[2]))) and

(not(zero(Cosines[3]))))

if ((Trend > 337.5) or (Trend <= 22.5))

then gotoxy(round(XVal/8),round((YVal-15)*3/25));

if ((Trend > 22.5) and (Trend <= 67.5))

then gotoxy(round((XVal+30)/8),round((Yval-15)*3/25));

then gotoxy(round((XVal+30)/8),round(YVal*3/25));

then gotoxy(round((XVal+30)/8),round((Yval+15)*3/25));

then gotoxy(round(XVal/8),round((YVal+15)*3/25));

then gotoxy(round((XVal-30)/8),round((Yval+15)*3/25));

then gotoxy(round((XVal-30)/8),round(YVal*3/25));

then gotoxy(round((XVal-30)/8),round((Yval-15)*3/25));

writeln(chr(232),' = 0');

if ((Count = (Intervals + 1)) and (not(Zero(Cosines[1]))) and

(not(Zero(Cosines[2]))) and (not(Zero(Cosines[3]))))

if ((Trend > 337.5) or (Trend <= 22.5))

then gotoxy(round(XVal/8),round((YVal-15)*3/25));

then gotoxy(round((XVal+30)/8),round((Yval-15)*3/25));

then gotoxy(round((XVal+30)/8),round(YVal*3/25));

then gotoxy(round((XVal+30)/8),round((Yval+15)*3/25));

then gotoxy(round(XVal/8),round((YVal+15)*3/25));

then gotoxy(round((XVal-30)/8),round((Yval+15)*3/25));

then gotoxy(round((XVal-30)/8),round(YVal*3/25));

then gotoxy(round((XVal-30)/8),round((Yval-15)*3/25));

writeln(chr(232),' = 1');

{ Main menu option # 5 - Display a numerical listing of the results }

procedure W riteList(Intervals : integer; Sigma1, Sigma3, Trend, Plunge : real;

SlipAngle, Ratios : linarray51; Sigma : chararray3);

{ Displays a numerical listing of the results }

label 1;

Count, IntPlunge, IntTrend, Page : integer;

Deviator, Step, Sigma2, Pages, Phi : real;

Sigma2 := Sigma3;

Pages := ((Intervals + 1.0) / 12.0);

if ((Pages - trunc(Pages)) > 0.00001) then Pages := Pages + 1.0;

for Page := 0 to (trunc(Pages) - 1) do

clrscr;

writeln;

textcolor(12);

writeln(' Slip Vector Plotting Program Results');

textcolor(11);

writeln;

write(' Data for a plane with a normal oriented at : ');

writeln(IntPlunge:2,' / ',IntTrend:3);

writeln;

write(' Sigma 1 = ',Sigma[1],' Sigma 2 = ',Sigma[2]);

writeln(' Sigma 3 = ',Sigma[3]);

writeln;

writeln(' Sigma 1 Sigma 2 Sigma 3 Phi Pitch Shear/Normal');

writeln;

for Count := ((Page * 12) + 1) to ((Page * 12) + 12) do

if (Count > (Intervals + 1)) then goto 1;

write(Sigma1:6:2,' ',Sigma2:6:2,' ',Sigma3:6:2,' ');

writeln(Phi:6:2,' ',SlipAngle[Count]:6:2,' ',Ratios[Count]:6:2);

1: end;

gotoxy(2,24);

write('Press any ');

textcolor(12);

write('key');

textcolor(11);

write(' to continue...');

repeat until (keypressed);

{ Main menu option # 6 - Procedures to print the results }

procedure AskDiskFileName(var FileName : String12);

{ Asks for the name of a disk data file }

FileLength, Count : integer;

Extension : boolean;

Key : char;

repeat

clrscr;

textcolor(12);

DrawBox(25,1,56,3);

textcolor(11);

gotoxy(27,2);

gotoxy(2,10);

write('Enter a filename (');

textcolor(12);

write('.DAT');

textcolor(11);

write(') : ');

textcolor(12);

readln(FileName);

textcolor(11);

FileLength := length(FileName);

Extension := false;

for Count := 1 to FileLength do

if (copy(FileName,Count,1) = '.')

Extension := true;

FileName := FileName + '.dat';

if ((Exists(FileName)) or (Extension))

if (Extension)

ShowError(9)

ShowError(10);

until ((not(Exists(FileName))) and (not(Extension)));

procedure AskAcadFileName(var FileName : String12);

{ Asks for the name of a disk data file }

FileLength, Count : integer;

Extension : boolean;

Key : char;

repeat

clrscr;

textcolor(12);

DrawBox(25,1,56,3);

textcolor(11);

gotoxy(27,2);

gotoxy(2,10);

write('Enter a filename (');

textcolor(12);

write('.DXF');

textcolor(11);

write(') : ');

textcolor(12);

readln(FileName);

textcolor(11);

FileLength := length(FileName);

Extension := false;

for Count := 1 to FileLength do

if (copy(FileName,Count,1) = '.')

Extension := true;

FileName := FileName + '.dxf';

if ((Exists(FileName)) or (Extension))

if (Extension)

ShowError(9)

ShowError(10);

until ((not(Exists(FileName))) and (not(Extension)));

procedure PrintMenuChoice(var Choice : integer);

{ Returns the number of the print menu item selected }

Key : char;

repeat

read(kbd,Key);

Choice := ord(Key) - 48;

if (not(Choice in [1..4]))

then Beep;

until (Choice in [1..4]);

procedure PrintMenu(var Choice : integer);

{ Displays the print menu }

Count : integer;

clrscr;

textcolor(12);

DrawBox(25,2,56,4);

textcolor(11);

gotoxy(27,3);

gotoxy(2,10);

writeln('What do you wish to do ?');

writeln;

textcolor(12);

write(' ',Count);

textcolor(11);

case (Count) of

1 : writeln(' - Write the numerical results to a disk file');

2 : writeln(' - Create a DXFile of the stereonet for AutoCAD');

3 : writeln(' - Create a DXFile of the graph for AutoCAD');

4 : writeln(' - Return to the main menu');

writeln;

write(' Press the ');

textcolor(12);

write('number');

textcolor(11);

write(' of you choice...');

MainMenuChoice(Choice);

{ Procedure to write the results to a disk file }

procedure WriteFile(Intervals : integer; Sigma1, Sigma3, Mu, Cohesion, Trend,

Plunge : real; SlipAngle, Ratios : linarray51;

FileName : string12; Sigma : chararray3);

{ W rites a disk data file of the results }

Count, IntPlunge, IntTrend : integer;

Deviator, Phi, Step, Sigma2 : real;

DataFile : text;

assign(DataFile,FileName);

rewrite(DataFile);

writeln(DataFile,' ');

writeln(DataFile,' Slip Vector Plotting Program');

write(DataFile,' Data for a plane with a normal oriented at : ');

writeln(DataFile,IntPlunge:2,' / ',IntTrend:3);

write(DataFile,' Sigma 1 = ',Sigma[1],' Sigma 1 = ',Sigma1:5:2);

writeln(DataFile,' Coefficient of friction = ',Mu:5:2);

write(DataFile,' Sigma 2 = ',Sigma[2],' Sigma 3 = ',Sigma3:5:2);

writeln(DataFile,' Cohesion = ',Cohesion:5:2);

write(DataFile,' Sigma 3 = ',Sigma[3]);

write(DataFile,' Sigma 1 Sigma 2 Sigma 3 Phi');

writeln(DataFile,' Pitch Stress ratio');

Sigma2 := Sigma3;

write(DataFile,Sigma1:6:2,' ',Sigma2:6:2,' ',Sigma3:6:2,' ');

write(DataFile,Phi:6:2,' ',SlipAngle[Count]:6:2,' ');

writeln(DataFile,Ratios[Count]:6:2);

close(DataFile);

{ AutoCAD DXFile creation procedures }

{$I DXF.INC}

{ Main Program }

QuitIt := false;

IsThereData := false;

IntroPage;

repeat

MainMenu(Choice);

case (Choice) of

1 : HelpPages;

2 : begin

AskData(Sigma,Sigma1,Sigma3,Mu,Cohesion,Plunge,Trend,DMin,DMax,Intervals);

Working;

DirCosines(Plunge,Trend,Dmin,DMax,Cosines);

Cauchy(Intervals,Sigma1,Sigma3,Cosines,Stresses);

CalculateStresses(Intervals,Mu,Cohesion,Trend,Plunge,Cosines,

Stresses,SlipVector,DMin,DMax,SlipAngle,Ratios);

Cursor(true);

IsThereData := true;

3 : begin

if (IsThereData)

AskCutOff(CutOff);

DrawGraphAxes(CutOff,Trend,Plunge,Sigma);

PlotPoints(Intervals,CutOff,Sigma1,Sigma3,Ratios,SlipAngle);

EndGraphics;

ShowError(11);

4 : begin

if (IsThereData)

AskCutOff(CutOff);

DrawStereonet(CutOff,Plunge,Trend,Sigma);

DrawFaultPlane(Plunge,Trend,90.0);

DataPlot(Intervals,CutOff,Trend,Cosines,SlipVector,Ratios);

EndGraphics;

ShowError(11);

5 : begin

if (IsThereData)

WriteList(Intervals,Sigma1,Sigma3,Trend,

Plunge,SlipAngle,Ratios,Sigma)

ShowError(11);

6 : begin

if (IsThereData)

Printmenu(Choice);

case (Choice) of

1 : begin

AskDiskFileName(FileName);

Writing(FileName);

WriteFile(Intervals,Sigma1,Sigma3,Mu,Cohesion,Trend,

Plunge,SlipAngle,Ratios,FileName,Sigma);

Cursor(true);

2 : begin

AskCutOff(CutOff);

AskAcadFileName(FileName);

CreatePage(FileName);

DXFileNet(Intervals,CutOff,Plunge,Trend,Cosines,

SlipVector,Ratios,FileName,Sigma);

Cursor(true);

3 : begin

AskCutOff(CutOff);

AskAcadFileName(FileName);

CreatePage(FileName);

DXFileGraph(Intervals,CutOff,Plunge,Trend,Sigma1,

Sigma3,Ratios,SlipAngle,FileName);

Cursor(true);

4 : {Continue}

ShowError(11);

7 : QuitIt := true;

Until (QuitIt);

ExitPage;

Listing of the include file DXF.INC which creates AutoCAD DXF files.

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * }

{ DXF.INC }

{ Procedures to create AutoCAD DXFiles for SLIP.PAS }

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * }

{ Procedures available: }

{ DXFileNet - Creates an AutoCAD DXfile for the stereonet }

{ DXFileGraph - Creates an AutoCAD DXFile for the graph }

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * }

procedure DXFileNet(Intervals : integer; CutOff, Plunge, Trend : real;

Cosines : linarray3; SlipVector : linarray2d;

Ratios : linarray51; FileName : string12;

Sigma : chararray3);

{ Creates an AutoCAD DXFile for the stereonet }

Strike, Dip, ApparentDip, Theta,

Distance, Beta, XVAl, YVAl, ZeroTrend : real;

DataFile : text;

rewrite(DataFile);

writeln(DataFile,'0');

writeln(DataFile,'SECTION');

writeln(DataFile,'HEADER');

writeln(DataFile,'$TEXTSTYLE');

writeln(DataFile,'SIMPLEX');

writeln(DataFile,'ENDSEC');

writeln(DataFile,'ENTITIES');

writeln(DataFile,'CIRCLE');

writeln(DataFile,'7.500');

writeln(DataFile,'LINE');

writeln(DataFile,'TEXT');

writeln(DataFile,'N');

writeln(DataFile,'Slip Vector');

writeln(DataFile,'Plotting Program');

writeln(DataFile,'Fault Plane Normal');

writeln(DataFile,'at ',IntPlunge:2,' / ',IntTrend:3);

writeln(DataFile,'Lower-Hemisphere');

writeln(DataFile,'Stereographic');

writeln(DataFile,'Projection');

writeln(DataFile,'Sigma 1 = ',Sigma[1]);

writeln(DataFile,' = Slip');

writeln(DataFile,' = Locked');

writeln(DataFile,'Cutoff Value = ',CutOff:4:2);

if (Plunge = 0.0)

then Plunge := Plunge + 0.0001;

Dip := DegToRad(90.0 - Plunge);

then Strike := Trend - 90.0

else Strike := Trend + 90.0;

Beta := DegToRad(Count / 10.0);

Distance := 4.000 * Tan((PI / 4.0) - (ApparentDip / 2.0));

if (Distance < 0.05)

Distance := (RadToDeg(Beta) * (2.0 / 45.0));

Beta := (((5.0 * PI) / 2.0) - Beta);

if (Beta >= (2.0 * PI))

if (Beta < 0.0)

if (Beta >= (2.0 * PI))

if (Beta < 0.0)

Beta := (((5.0 * PI) / 2.0) - Beta);

if (Beta >= (2.0 * PI))

if (Beta < 0.0)

XVal := (7.500 + (Distance * cos(Beta)));

YVal := (4.500 + (Distance * sin(Beta)));

writeln(DataFile,'POINT');

writeln(DataFile,XVAL);

writeln(DataFile,YVAL);

Distance := (4.000 * Tan((PI / 4.0) - (DegToRad(SlipVector[1,Count]) / 2.0)));

if (SlipVector[1,Count] = 0.0)

ZeroTrend := (SlipVector[2,Count] + 180.0);

XVal := (7.500 + (Distance * cos(Theta)));

YVal := (4.500 + (Distance * sin(Theta)));

then writeln(DataFile,'0.150')

else writeln(DataFile,'0.050');

Theta := (((5.0 * PI) / 2.0) - DegToRad(SlipVector[2,Count]));

if ((Count = 1) and (not(Zero(Cosines[1]))) and (not(Zero(Cosines[2])))

and (not(Zero(Cosines[3]))))

if ((Trend > 337.5) or (Trend <= 22.5))

writeln(DataFile,(XVAL));

writeln(DataFile,(YVAL+0.50));

writeln(DataFile,(XVAL+0.50));

writeln(DataFile,(YVAL));

writeln(DataFile,(YVAL-0.50));

writeln(DataFile,(XVAL-0.50));

writeln(DataFile,'Phi = 0');

if ((Count = (Intervals + 1)) and (not(Zero(Cosines[1]))) and

(not(Zero(Cosines[2]))) and (not(Zero(Cosines[3]))))

if ((Trend > 337.5) or (Trend <= 22.5))

writeln(DataFile,'Phi = 1');

writeln(DataFile,'EOF');

close(DataFile);

procedure DXFileGraph(Intervals : integer; CutOff, Plunge, Trend, Sigma1, Sigma3 : real;

Ratios, SlipAngle : linarray51; FileName : string12);

{ Creates an AutoCAD DXFile for the graph }

Deviator, Phi, Step, Sigma2, XVal, YVal : real;

DataFile : text;

rewrite(DataFile);

writeln(DataFile,'HEADER');

writeln(DataFile,'$TEXTSTYLE');

writeln(DataFile,'SIMPLEX');

writeln(DataFile,'ENTITIES');

XVal := ((2.0 * Count) + 3.000);

writeln(DataFile,(XVal - 0.20));

writeln(DataFile,(Count * 0.2):2:1);

YVal := ((1.25 * Count) + 2.000);

writeln(DataFile,(YVal - 0.1));

case (Count) of

0 : writeln(DataFile,'-90');

1 : writeln(DataFile,'-45');

2 : writeln(DataFile,' 0');

writeln(DataFile,'Slip Vector Plotting Program');

writeln(DataFile,'Fault Plane Normal at ',IntPlunge:2,' / ',IntTrend:3);

writeln(DataFile,'Cutoff Value = ',CutOff:4:2);

writeln(DataFile,'Pitch');

writeln(DataFile,'Fault Plane');

writeln(DataFile,' = Slip');

writeln(DataFile,' = Locked');

writeln(DataFile,'Phi');

Sigma2 := Sigma3;

XVal := (3.000 + (Phi * 10.0));

YVal := (4.500 + ((SlipAngle[Count] * 5.0) / 180.0));

if (abs(abs(SlipAngle[Count]) - 90.0) < 0.0001)

YVal := (4.500 + ((SlipAngle[Count] * 5.0) / 180.0));

YVal := (4.500 - ((SlipAngle[Count] * 5.0) / 180.0));

writeln(DataFile,'EOF');

close(DataFile);

Listing of the help file SLIP.TXT called by SLIP.PAS, the program listed above.

SLIP.TXT - A 6 page help file for the program SLIP.PAS

An asterisk before a line causes it to be highlighted when the help option

is chosen in the program. The first five lines of this file are not read.

* Slip Vector Plotting Program

This is a program to calculate the slip vector and the stress ratios

acting upon a fault plane of any arbitrary orientation with a varying

magnitude of F2 given fixed magnitudes for F1 and F3 where the principal

stress axes may be oriented either north/south, east/west, or vertical at

your choosing. The results of this program are then displayed as a graph

of the ratio of the principal stresses versus the pitch of the slip vector

from the strike of the fault plane, as a lower-hemisphere stereographic

projection of the fault plane and the slip vectors resulting from differing

values of F2, or as a numerical listing of the values of F1, F2, and F3 and

the pitch of the slip vectors and the stress ratios which are associated

with them. The stress ratios are defined as [Fs / (:Fn + C)] where Fs is

the shear stress, Fn is the normal stress, : is the coefficient of friction,

and C is the cohesion.

The idea for this program came from a paper in volume 96 of Geological

Magazine by M. H. P. Bott entitled "The Mechanics of Oblique Slip Faulting"

published in 1959. Bott showed that the position of the slip vector in the

fault plane is dependant upon the ratios of the three principal stresses.

When you begin program execution, you will be asked to provide a value

for F1 and F3. The program will accept any positive real numbers as valid

values provided, of course, that the value for F1 is greater than that for

F3. A value for the coefficient of friction (:) and the cohesion must then

be supplied. Next, the values of the plunge and trend of the normal to the

fault plane you wish to examine must be entered. The plunge may be any real

number between 0 and 89.9 inclusive and the trend may be any real number

between 0 and 359.9 inclusive. The program will then ask for the number of

intervals of F2 you wish to examine as F2 varies between F3 and F1 in

however many increments you specify. The program will accept any integer

between 2 and 50 inclusive for this value.

Although the program will accept any real numbers greater than or equal

to zero for the values of F1, F3, :, and C, it must be kept in mind that

there are many values which could be entered which may not have any real

geological significance. Always try to keep in mind as to what might be

realistic values for the deviatoric stress (F1 - F3) and for the isotropic

stress [(F1 + F2 + F3) / 3.0].

After this program has run, you may examine the data in several ways.

First, you may examine a graph plotting M versus the pitch of the slip

vector from the strike direction. M is defined as [(F2 - F3) / (F1 - F3)]

and the pitch varies from -90.0 to +90.0 degrees as an angle from the strike

direction which is always between 0.0 and 180.0 degrees inclusive. Second,

a lower-hemisphere stereograph projection may be examined which plots the

fault plane and the positions of the different slip vectors on it for the

differing values of F2. Third, the numerical values of F1, F2, and F3 may

be written along with the values for the pitch of the slip vectors and the

ratio of the shear to normal stresses associated with them. This data may

also be printed by writing it to a disk data file and then using the DOS

command:

* PRINT filename.ext

The graph and the stereonet may also be written to an AutoCAD drawing

interchange file (DXF).

To create and print an AutoCAD drawing interchange file, first write the

data to a file with the extension .DXF and then read that file into AutoCAD

using the DXFIN command. Once in AutoCAD, changes may be made if desired and

the drawing may then be plotted using a laser printer or pen plotter.

Before creating the graph or stereonet plots, the program will ask for a

cutoff value for the shear to normal stress ratio. When a value greater than

0.0 is specified, any slip vector with a stress ratio above the cutoff will

be plotted as a circle and any slip vector with a value below the cutoff will be

plotted as a square. The rational for doing this is that faulting will occur

only when the stress ratio exceeds a certain value. When a geologically

reasonable value is entered, it may be assumed that any vector plotting as

a circle may initiate slip while any vector plotting as a square will leave

the fault remaining locked.

The main purpose of this program is to generate a data set of known fault

orientations and their slip directions given a specified stress field. This

data will then be used to test and evaluate different computer methods for

determining principal stress orientations from faults and their slip vectors.

This program was written by Steven H. Schimmrich in Turbo Pascal (Borland,

Inc. - registered trademark) for an IBM AT personal computer with a numeric

coprocessor, 512 K memory, and color graphics capabilities. The program may

not run properly on other machines. Version 1.0 of this program was written

in April of 1988, version 2.0 in July of 1988. This help file that you are

reading is called SLIP.TXT and must be on the same disk as the main program

SLIP.PAS is. The compiled version of this program is called SLIP.COM and may

be run from the DOS level by simply typing SLIP at the DOS prompt.

If at any time during the program execution it becomes "stuck" or the

keyboard freezes, pressing the Ctrl key and the C letter key at the same

time will usually return you to the DOS level. If that fails try pressing

the Ctrl, Alt, and Del keys all at the same time to reboot the computer.

* SLIP.PAS

* Steven H. Schimmrich

APPENDIX D

FAULT PLANE PLOTTING PROGRAM

Complete listing of the fault plane plotting program discussed in chapters six and seven. The

program FPLANE.PAS is written in Turbo Pascal version 3.01 for an IBM PC or compatible computer.

program FPlane;

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *}

{ This is a program to create an AutoCAD script file of planes and }

{ their associated slip directions for plotting on a lower-hemisphere }

{ equal-angle stereographic projection. }

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *}

{ FPLANE.PAS - Version 1.0 }

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *}

{ Initializations }

RegisterList =

record

AX,BX,CX,DX,BP,SI,DI,DS,ES,Flags : integer;

RArr = array[1..100] of real;

String25 = string[25];

Plunge, Trend, Pitch : RArr;

Number : integer;

FileName : String25;

{General program functions }

function Exists(FileName : String25): boolean;

{ Checks to see if a file exists on the disk }

Name : file;

Assign(Name,FileName);

Reset(Name);

Exists := (IOResult = 0);

procedure Cursor(On : boolean);

{ Turns cursor on and off }

Register : RegisterList;

if (On)

if (mem[0:$449] = 7)

Register.CX := $0C0D

Register.CX := $0607

Register.CX := $2000;

Register.AX := $0100;

intr($10,Register);

function Zero(Value : real): boolean;

{ Checks to see if a value is essentially 0.0 }

if (abs(Value) < 0.00001)

then Zero := true

else Zero := false;

function DegToRad(DegreeMeasure : real): real;

DegToRad := ((DegreeMeasure * PI) / 180.0);

function RadToDeg(RadianMeasure : real): real;

RadToDeg := ((RadianMeasure * 180.0) / PI);

function Tan(Angle : real): real;

{ Returns the tangent of an angle }

Tan := sin(Angle) / cos(Angle);

X, Y : real;

if (AValue = 0.0)

then ArcCos := 0.0

then ArcCos := PI

if (X > 0.0)

then ArcCos := Y

{ General program procedures }

procedure DrawBox(ULX, ULY, LRX, LRY : integer);

{ Draws a box around text in text mode }

X, Y, XDistance : integer;

gotoxy(ULX,ULY);

write(#201);

XDistance := LRX - ULX - 1;

write(#205);

write(#187);

for Y := (ULY + 1) to (LRY - 1) do

gotoxy(LRX,Y);

write(#186);

gotoxy(ULX,Y);

write(#186);

gotoxy(ULX,LRY);

write(#200);

write(#205);

write(#188);

procedure DisplayPage;

{ Displays an introduction page }

textmode;

clrscr;

cursor(false);

textcolor(12);

DrawBox(20,9,61,18);

textcolor(11);

gotoxy(27,10);

writeln('Fault Plane Plotting Program');

gotoxy(32,12);

writeln('Program written by');

gotoxy(31,14);

gotoxy(24,15);

writeln('Department of Geological Sciences');

gotoxy(22,16);

writeln('State University of New York at Albany');

gotoxy(30,17);

writeln('Albany, New York 12222');

delay(5000);

cursor(true);

clrscr;

procedure IntroPage;

{ Displays a short introduction to the program }

Key : char;

gotoxy(27,2);

textcolor(12);

textcolor(11);

writeln;

writeln(' This is a quick and dirty program to plot fault planes and their associated');

writeln(' slip directions onto an equal-angle stereographic projection using an AutoCAD');

writeln(' SCRipt file. This program was written because the RockWare stereonet program');

writeln(' on this computer can not plot planes as great circles easily.');

writeln(' This program first asks for a filename of a disk file containing the fault');

writeln(' data. This data must have the following format : ');

writeln;

textcolor(12);

writeln(' 25 125 13');

textcolor(11);

writeln;

writeln(' where 25 is the plunge and 125 is the trend of the fault plane normal and 13');

writeln(' is the pitch of the slip vector within the fault plane. The pitch is the angle');

writeln(' between the strike (the trend of the normal + 90 degrees) and the slip vector');

writeln(' which varies from -90 to 90 degrees. If you do not wish to plot the slip');

writeln(' vectors, enter a 99 for the pitch. Entering a 999 for the pitch will treat');

writeln(' the plunge and trend values as an axis and plot a circle.');

write(' The program will then create an AutoCAD file named FPLANE.SCR ');

writeln('overwriting');

writeln(' any old versions of FPLANE.SCR if they exist on the disk. The file may then');

writeln(' be read into AutoCAD and plotted using the laser printer. If any text is');

writeln(' desired, it may be added while in AutoCAD.');

writeln;

gotoxy(2,25);

write('Press any ');

textcolor(12);

write('key');

textcolor(11);

write(' to continue...');

read(kbd,Key);

clrscr;

procedure AskFileName(var FileName : String25);

{ Asks for the name of the data file }

gotoxy(27,2);

textcolor(12);

textcolor(11);

writeln;

writeln(' The program now needs the name of the disk file');

writeln(' containing the fault normals and slip directions.');

repeat

writeln;

write(' Enter the name of the data file : ');

textcolor(12);

readln(FileName);

textcolor(11);

if (not(Exists(FileName)))

sound(880);

delay(50);

nosound;

writeln;

textcolor(12);

writeln(' * ERROR * That file does not exist on this disk');

textcolor(11);

until (Exists(FileName));

clrscr;

procedure ReadData(FileName : String25; var Number : integer;

var Plunge, Trend, Pitch : RArr);

{ Reads in the data from a disk file }

DataFile : text;

Number := 0;

gotoxy(27,2);

textcolor(12);

textcolor(11);

gotoxy(2,8);

write('Creating AutoCAD file...');

reset(DataFile);

while (not(EOF(DataFile))) do

Number := Number + 1;

readln(DataFile,Plunge[Number],Trend[Number],Pitch[Number]);

close(DataFile);

procedure CreatePlot(var Number : integer; var Plunge, Trend, Pitch : RArr);

{ Creates the AutoCAD plot of data }

label 1;

label 2;

ACADFILE = 'FPLANE.SCR';

Strike, Dip, ApparentDip, Theta,

Distance, Beta, XVal, YVal, XVal1,

Yval1, XVal2, YVal2, XVal3, YVal3,

ZeroTrend, SlipTrend, SlipPlunge : real;

Count, Counter : integer;

DataFile : text;

assign(DataFile,ACADFILE);

rewrite(DataFile);

writeln(DataFile,'7.500,4.500');

writeln(DataFile,'MIDDLE');

writeln(DataFile,'N');

for Count := 1 to Number do

if (Pitch[Count] = 999.0) then goto 1;

if (Plunge[Count] = 0.0)

then Plunge[Count] := Plunge[Count] + 0.0001;

Dip := DegToRad(90.0 - Plunge[Count]);

if ((Trend[Count] >= 90.0) and (Trend[Count] < 270.0))

then Strike := Trend[Count] - 90.0

else Strike := Trend[Count] + 90.0;

for Counter := 0 to 2 do

Beta := DegToRad(Counter * 90.0);

Distance := 4.000 * Tan((PI / 4.0) - (ApparentDip / 2.0));

if (Distance < 0.05)

Distance := (RadToDeg(Beta) * (2.0 / 45.0));

Beta := (((5.0 * PI) / 2.0) - Beta);

if (Beta >= (2.0 * PI))

if (Beta < 0.0)

if ((Trend[Count] >= 90.0) and (Trend[Count] < 270.0))

if (Beta >= (2.0 * PI))

if (Beta < 0.0)

Beta := (((5.0 * PI) / 2.0) - Beta);

if (Beta >= (2.0 * PI))

if (Beta < 0.0)

XVal := (7.500 + (Distance * cos(Beta)));

YVal := (4.500 + (Distance * sin(Beta)));

case (Counter) of

0 : begin

XVal1 := XVal;

YVal1 := YVal;

1 : begin

XVal2 := XVal;

YVal2 := YVal;

2 : begin

XVal3 := XVal;

YVal3 := YVal;

if ((abs(XVal1 - XVal2) >= 0.01) or (abs(YVal1 - YVal2) >= 0.001))

writeln(DataFile,'ARC');

writeln(DataFile,XVal1:5:3,',',YVal1:5:3);

if (Pitch[Count] = 99.0) then goto 2;

if (not(Zero(Pitch[Count])))

then Beta := RadToDeg(arctan(cos(DegToRad(90 - Plunge[Count])) *

Tan(DegToRad(Pitch[Count]))))

else Beta := 0.0;

Strike := Trend[Count] + 90.0;

if (Strike > 360.0)

SlipTrend := Strike + Beta;

if (not(Zero(abs(Beta) - 90.0)))

then SlipPlunge := RadToDeg(ArcCos(cos(DegToRad(Pitch[Count])) /

cos(DegToRad(Beta))))

else SlipPlunge := 90.0 - Plunge[Count];

if ((Pitch[Count] < 0.0) or (Pitch[Count] = 90.0))

then SlipTrend := SlipTrend + 180.0;

if (Trend[Count] > 270)

then SlipTrend := SlipTrend + 180.0;

if (SlipTrend > 360.0)

then SlipTrend := SlipTrend - 360.0;

Distance := (4.000 * Tan((PI / 4.0) - (DegToRad(SlipPlunge) / 2.0)));

if (SlipPlunge = 0.0)

ZeroTrend := (SlipTrend + 180.0);

XVal := (7.500 - (Distance * cos(Theta)));

YVal := (4.500 - (Distance * sin(Theta)));

writeln(DataFile,XVAL:5:3,',',YVAL:5:3);

Theta := (((5.0 * PI) / 2.0) - DegToRad(SlipTrend));

if (Pitch[Count] = 90.0)

XVal := (7.500 - (Distance * cos(Theta)));

YVal := (4.500 - (Distance * sin(Theta)));

goto 2;

1: {continue};

Distance := (4.000 * Tan((PI / 4.0) - (DegToRad(Plunge[Count]) / 2.0)));

Theta := (((5.0 * PI) / 2.0) - DegToRad(Trend[Count]));

2: {continue};

writeln(DataFile,'REGEN');

close(DataFile);

gotoxy(2,12);

write('AutoCAD file ');

textcolor(12);

write('FPLANE.SCR');

textcolor(11);

write(' created...');

sound(880);

delay(50);

nosound;

delay(5000);

{ Main program }

DisplayPage;

IntroPage;

AskFileName(FileName);

ReadData(FileName,Number,Plunge,Trend,Pitch);

CreatePlot(Number,Plunge,Trend,Pitch);

DisplayPage;

A P P E N D IX E

V E C T O R A N G L E C A L C U L A T IO N P R O G R A M

C o m p le t e l i s t i n g o f th e v e c t o r a n g le c a l c u l a t i o n p ro g ra m d isc usse d in c h a p te r n in e . T h e

p r o g r a m A N G L E .P A S i s w r i t t e n i n T u r b o P a s c a l v e r s i o n 3 . 0 1 f o r a n I B M P C o r c o m p a t i b l e

c o m p u te r .

program Angle;

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * }

{ This is a program to calculate the angles between two vectors }

{ given their stereographic plunges and trends in degrees. }

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * }

{ ANGLE.PAS - Version 1.0 }

{ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * }

{ Initializations }

Plunge1, Plunge2, Trend1, Trend2 : integer;

Angle, DotProduct, X1, Y1, Z1, X2, Y2, Z2 : real;

Loop : boolean;

Key : char;

function DegtoRad(DegreeMeasure : real): real;

DegtoRad := ((DegreeMeasure * PI) / 180.0);

function RadtoDeg(RadianMeasure : real): real;

RadtoDeg := ((RadianMeasure * 180.0) / PI);

X, Y : real;

if (AValue = 0.0)

then ArcCos := 0.0

then ArcCos := PI

if (X > 0.0)

then ArcCos := Y

{ Main program }

Loop := true;

while (Loop) do

clrscr;

textcolor(12);

writeln(' Angle Calculation Program');

textcolor(14);

writeln;

writeln(' Given the plunge and trend values in degrees of two');

writeln('vectors, this program will calculate the angle between');

writeln('them in degrees.');

writeln;

write('Vector');

textcolor(12);

writeln(' 1');

textcolor(14);

writeln;

write(' Enter the plunge : ');

textcolor(12);

readln(Plunge1);

textcolor(14);

write(' Enter the trend : ');

textcolor(12);

readln(Trend1);

textcolor(14);

writeln;

write('Vector');

textcolor(12);

writeln(' 2');

textcolor(14);

writeln;

write(' Enter the plunge : ');

textcolor(12);

readln(Plunge2);

textcolor(14);

write(' Enter the trend : ');

textcolor(12);

readln(Trend2);

textcolor(14);

if ((Plunge1 > 90) or (Plunge1 < 0) or (Plunge2 > 90) or (Plunge2 < 0)

or (Trend1 > 360) or (Trend1 < 0) or (Trend2 > 360) or (Trend2 < 0))

writeln;

sound(880);

delay(100);

nosound;

textcolor(12);

write('ERROR');

textcolor(14);

writeln(' - An incorrect plunge or trend has been entered!');

X1 := (cos(DegtoRad(Plunge1)) * cos(DegtoRad(Trend1)));

Y1 := (sin(DegtoRad(Plunge1)));

Z1 := (cos(DegtoRad(Plunge1)) * sin(DegtoRad(Trend1)));

X2 := (cos(DegtoRad(Plunge2)) * cos(DegtoRad(Trend2)));

Y2 := (sin(DegtoRad(Plunge2)));

Z2 := (cos(DegtoRad(Plunge2)) * sin(DegtoRad(Trend2)));

DotProduct := ((X1 * X2) + (Y1 * Y2) + (Z1 * Z2));

Angle := (RadtoDeg(ArcCos(DotProduct)));

if (Angle > 90.0)

then Angle := 180.0 - Angle;

writeln;

write('The angle between vectors');

textcolor(12);

write(' 1');

textcolor(14);

write(' and');

textcolor(12);

write(' 2');

textcolor(14);

write(' is');

textcolor(12);

write(Angle:6:1);

textcolor(14);

writeln(' degrees.');

writeln;

write('Do another calculation (');

textcolor(12);

write('Y');

textcolor(14);

write(' or');

textcolor(12);

write(' N');

textcolor(14);

write(') ? ');

read(kbd,Key);

if (Key in ['y','Y'])

then Loop := true

else Loop := false;

clrscr;

R E F E R E N C E S

A le k s a n d ro w s k i , P . 1 9 8 5 . G ra p h i c a l d e t e rm in a t io n o f p r in c ip a l s t r e s s d i r e c t i o n s fo r

s l i c k e n s id e l i n e a t i o n p o p u la t i o n s : A n a t t e m p t t o m o d i f y A r th a u d 's m e th o d . J o u rn a l

o f S tru c tu ra l G e o lo g y . 7 : 7 3 -8 2 .

A l l m e n d i n g e r , R . W . 1 9 8 9 . P e r s o n a l c o m m u n i c a t i o n w i t h R i c h a r d A l l m e n d i n g e r o f C o r n e l l

U n iv e rs i t y .

A n d e rs o n , E . M . 1 9 5 1 . T h e D y n a m ics o f F a u l t in g a n d D y k e F o rm a t io n w i th A p p l ica t io n s to

B r i ta in . 2 n d e d i t i o n . O l iv e r a n d B o y d : E d i n b u rg h . 2 0 6 p a g e s .

A n g e l i e r , J . 1 9 7 5 . S u r l 'a n a l y s e d e m e s u re s r e c u e i l l i e s d a n s d e s s i t e s f a i l l é s : L 'u t i l i t é d 'u n e

c o n f r o n ta t i o n e n t r e l e s m é th o d e s d y n a m i q u e s e t c in è m a t i q u e s . C o m p te s R e n d u s

H e b d o m a d a ire s d e L 'A c a d é m ie d e s S c ie n c e s F ra n c e , S é r ie D . 2 8 1 :1 8 0 5 -1 8 0 8 .

A n g e l i e r , J . a n d M e c h le r , P . 1 9 7 7 . S u r u n e m é th o d e g ra p h iq u e d e re c h e rc h e d e s c o n s t r a in te s

é g a l e m e n t u t i l i s a b l e e n t e c t o n iq u e e t e n s é i s m o l o g i e : L a m é t h o d e d e s d i è d r e s d r o i t s .

B u l le t in d e la S o c ié té G é o lo g iq u e d e F r a n c e . 1 9 : 1 3 0 9 -1 3 1 8 .

A n g e l i e r , J . 1 9 7 9 . D e t e r m i n a t i o n o f t h e m e a n p r i n c i p a l d i r e c t i o n s o f s t re s s e s f o r a g i v e n f a u l t

p o p u la t i o n . T e c to n o p h y s ic s . 5 6 :T 1 7 -T 2 6 .

A n g e l i e r , J . a n d G o g u e l , J . 1 9 7 9 . S u r u n e m é t h o d e s im p le d e d é te rm in a t io n d e s a x e s

p r i n c ip a u x d e s c o n s t r a in te s p o u r u n e p o p u la t i o n d e fa i l l e s . C o m p te s R e n d u s

H e b d o m a d a ire s d e L 'A c a d é m ie d e s S c ie n c e s F ra n c e , S é r ie D . 2 8 8 : 3 0 7 -3 1 0 .

A n g e l i e r , J . a n d M a n o u ss i s , S . 1 9 8 0 . C la s s i f i c a t i o n a u t o m a t iq u e e t d i s t i n c t io n d e s p h a s e s

s u p e rp o s é e s e n te c to n iq u e d e fa i l l e s . C o m p te s R e n d u s H e b d o m a d a ire s d e L 'A c a d é m ie

d e s S c ie n c e s F ra n c e , S é r ie D . 2 9 0 :6 5 1 -6 5 4 .

A n g e l i e r , J . , T a r a n t o l a , A . , V a l e t t e , B . , a n d M a n o u s s i s , S . 1 9 8 2 . In v e r s i o n o f f i e l d d a t a i n

fa u l t t e c to n i c s t o o b t a in t h e r e g io n a l s t r e s s . I . S i n g l e p h a s e f a u l t p o p u la t i o n s : A n e w

m e th o d o f c o m p u t i n g th e s t r e s s t e n s o r . G e o p h ys ic a l J o u rn a l o f th e R o y a l A s tro n o m ic a l

S o c ie ty . 6 9 : 6 0 7 -6 2 1 .

A n g e l i e r , J . 1 9 8 4 . T e c to n ic a n a ly s i s o f fa u l t s l i p d a ta s e t s . J o u r n a l o f G e o p h y s ic a l R e se a rc h .

8 9 : 5 8 3 5 -5 8 4 8 .

A n g e l i e r , J . , C o l l e t t a , B . , a n d A n d e r s o n , R . E . 1 9 8 5 . N e o g e n e p a l e o s t r e s s c h a n g e s i n t h e B a s i n

a n d R a n g e : A c a s e s tu d y a t H o o v e r D a m , N e v a d a -A r i z o n a . G e o lo g ic a l S o c ie ty o f

A m e r ic a B u l le t in . 9 6 :3 4 7 -3 6 1 .

A n g e l i e r , J . 1 9 8 9 . F r o m o r i e n t a t i o n t o m a g n i t u d e s i n p a l e o s t r e s s d e t e r m i n a t i o n s u s i n g f a u l t

s l i p d a ta . J o u rn a l o f S tru c tu ra l G e o lo g y . 1 1 : 3 7 -5 0 .

A rm i jo , R . a n d C i s t e rn a s , A . 1 9 7 8 . U n p r o b l è m e in v e rs e e n m ic ro te c to n iq u e c a s s a n te .

C o m p tes R en du s H e b d o m a d a ire s d e L 'A c a d é m ie d e s S c ie n c e s F ra n c e , S é r ie D . 2 8 7 :5 9 5 -

5 9 8 .

A rm i jo , R . , C a re y , E . , a n d C i s te rn a s , A . 1 9 8 2 . T h e in v e r s e p ro b le m in m ic ro t e c t o n ic s a n d th e

s e p a ra t i o n o f t e c to n ic p h a s e s . T e c to n o p h y s ic s . 8 2 :1 4 5 -1 6 0 .

A t h a u d , F . 1 9 6 9 . M é t h o d e d e d é t e r m i n a t i o n g r a p h i q u e d e s d i r e c t i o n s d e r a c c o u r c i s s e m e n t ,

d 'a l l o n g e m e n t e t i n te rm é d ia i r e d 'u n e p o p u la t i o n d e fa i l l e s . B u l le t in d e la S o c ié té

G é o lo g iq u e d e F r a n c e . 1 1 : 7 2 9 -7 3 7 .

A r th a u d , F . a n d M a t t a u e r , M . 1 9 6 9 . E x a m p le s d e s ty lo l i t e s d 'o r ig in e t e c t o n iq u e d a n s l e

L a n g u e d o c , l e u r s re l a t i o n s a v e c l a t e c t o n i q u e c a s s a n t e . B u l le t in d e la S o c ié té

G é o lo g iq u e d e F r a n c e . 1 1 : 7 3 8 -7 4 4 .

A y d in , A . 1 9 7 7 . F a u l t in g in S a n d s to n e . P h .D . d i s s e r t a t i o n . S ta n fo rd U n iv e rs i t y .

A y d i n , A . 1 9 8 0 . D e t e r m i n a t i o n o f t h e o r i e n t a t i o n o f t h e p r i n c i p a l s t r e s s e s f ro m t h r e e o r m o r e

s e t s o f c o n te m p o ra n e o u s fa u l t s (a b s t r a c t ) . E O S : T ra n sa c t io n s o f th e A m e ric a n

G e o p h ys ic a l U n io n . 6 1 :1 1 1 7 .

A y d in , A . a n d R e c h e s , Z . 1 9 8 2 . N u m b e r a n d o r i e n t a t i o n o f f a u l t s e t s i n t h e f i e l d a n d i n

e x p e r i m e n ts . G e o lo g y . 1 0 : 1 0 7 -1 1 2 .

B a h a t , D . a n d R a b i n o v i c h , A . 1 9 8 8 . P a l e o s t r e s s d e t e r m i n a t i o n i n a r o c k b y a f ra c t o g r a p h i c

m e th o d . J o u rn a l o f S tru c tu ra l G e o lo g y . 1 0 : 1 9 3 -1 9 9 .

B a r t o n , N . a n d C h o u b e y , V . 1 9 7 7 . T h e s h e a r s t r e n g th o f ro c k s in th e o ry a n d p ra c t i c e . R o c k

M e c h a n ic s . 1 0 :1 -5 4 .

B e r g e r , A . R . 1 9 7 1 . D y n a m ic a n a ly s i s u s i n g d y k e s w i th o b l iq u e i n t e rn a l fo l i a t io n s .

G e o lo g ica l S o c ie ty o f A m e r ic a B u l le t in . 8 2 : 7 8 1 -7 8 6 .

B i l l i n g s , M . P . 1 9 7 2 . S tru c tu ra l G e o lo g y . 3 rd e d i t i o n . P r e n t i c e -H a l l : E n g le w o o d C l i f f s , N J .

6 0 6 p a g e s .

B o r ra d a i l e , G . J . 1 9 8 6 . T h e in te rn a l t e c to n i c f a b r i c o f m in o r in t ru s io n s a n d th e i r p o t e n t i a l a s

re g io n a l p a le o s t r e s s in d ic a to rs . G e o lo g ic a l M a g a z in e . 1 2 3 : 6 6 5 -6 7 1 .

B o t t , M . H . P . 1 9 5 9 . T h e m e c h a n ic s o f o b l i q u e -s l i p fa u l t in g . G e o lo g ic a l M a g a z in e .

9 6 : 1 0 9 -1 1 7 .

B r a c e , W . F . a n d K o h l s t e d t , D . L . 1 9 8 0 . L i m i t s o n l i t h o s p h e r i c s t r e s s i m p o s e d b y l a b o r a t o r y

e x p e r i m e n ts . J o u r n a l o f G e o p h y s ic a l R e se a rc h . 8 5 : 6 2 4 8 -6 2 5 2 .

B ro w n , S . R . a n d S c h o lz , C . H . 1 9 8 5 . B ro a d b a n d w id th s tu d y o f t h e t o p o g ra p h y o f n a tu ra l ro c k

s u r f a c e s . J o u r n a l o f G e o p h y s ic a l R e se a rc h . 9 0 :1 2 ,5 7 5 -1 2 ,5 8 2 .

B u c h n e r , F . 1 9 8 1 . R h in e g ra b e n : H o r i z o n t a l s t y lo l i t e s in d i c a t in g s t r e s s r e g im e s o f e a r l i e r

s ta te s o f r i f t in g . T e c to n o p h y s ic s . 7 3 : 1 1 3 -1 1 8 .

B e u tn e r , E . C . 1 9 7 2 . R e v e r s e g ra v i t a t i o n a l m o v e m e n t o n e a r l i e r o v e r th ru s t s , L e m h i R a n g e ,

Id a h o . G e o lo g ica l S o c ie ty o f A m e r ic a B u l le t in . 8 3 : 8 3 9 -8 4 7

B y e r l e e , J . D . 1 9 6 8 . B r i t t l e -d u c t i l e t ra n s i t io n in ro c k s . J o u r n a l o f G e o p h y s ic a l R e se a rc h .

7 3 : 4 7 4 1 -4 7 5 0 .

B y e r l e e , J . D . 1 9 7 8 . F r i c t i o n o f ro c k s . P u r e a n d A p p l ie d G e o p h y s ic s . 1 1 6 :6 1 5 -6 2 6 .

B y e r l e e , J . D . , M ja c h k i n , V . , S u m m e rs , R . , a n d V o e v o d a , O . 1 9 7 8 . S t ru c tu re s d e v e l o p e d i n

fa u l t g o u g e d u r i n g s ta b le s l i d in g a n d s t i c k -s l i p . T e c to n o p h y s ic s . 4 4 :1 6 1 -1 7 1 .

C a p u to , M . a n d C a p u to , R . 1 9 8 8 . S t ru c tu ra l a n a ly s i s : N e w a n a ly t i c a l a p p ro a c h a n d

a p p l i c a t i o n s . A n n a le s T e c to n ic æ . 2 : 8 4 -8 9 .

C a re y , E . a n d B ru n ie r , B . 1 9 7 4 . A n a l y s e t h é o r iq u e e t n u m é r iq u e d 'u n m o d è l e m é c h a n i q u e

é lé m e n ta i r e a p p l i q u é à l 'é tu d e d 'u n e p o p u la t i o n d e fa i l l e s . C o m p te s R e n d u s

H e b d o m a d a ire s d e L 'A c a d é m ie d e s S c ie n c e s F ra n c e , S é r ie D . 2 7 9 : 8 9 1 -8 9 4 .

C a re y , E . 1 9 7 6 . A n a ly s e n u m é r iq u e d 'u n m o d è l e m é c h a n i q u e é lé m e n ta i r e a p p l iq u é à l 'é t u d e

d 'u n e p o p u l a t i o n d e f a i l l e s : C a l c u l d 'u n t e n s e u r m o y e n d e s c o n s t r a in t e s a p a r t i r d e

s t r i e s d e g l i s se m e n t . T h è s e 3 è m e c y c l e . U n iv e r s i t é d e P a r i s S u d .

C a r t e r , N . L . a n d R a l e i g h , C . B . 1 9 6 9 . P r in c ip a l s t r e s s d i re c t i o n s f ro m p l a s t i c f lo w in c ry s t a l s .

G e o lo g ica l S o c ie ty o f A m e r ic a B u l le t in . 8 0 : 1 2 3 1 -1 2 6 4 .

C é l é r i e r , B . 1 9 8 8 . H o w m u c h d o e s s l i p o n a r e a c t i v a t e d f a u l t p l a n e c o n s t r a i n t h e s t r e s s t e n s o r ?

T e c to n ic s . 7 : 1 2 5 7 -1 2 7 8 .

C h e e n e y , R . F . 1 9 8 3 . S ta t i s t ic a l M e th o d s in G e o lo g y . G e o rg e A l l e n & U n w in : L o n d o n . 1 6 9

p a g e s .

C o m p to n , R . R . 1 9 6 2 . M a n u a l o f F ie ld G e o lo g y . Jo h n W i l e y : N e w Y o rk , N Y . 3 7 8 p a g e s .

C o u lo m b , C . A . 1 7 7 6 . S u r u n e a p p l i c a t i o n d e s r è g le s d e m a x im is e t m in im is à q u e lq u e s

p ro b lè m e s d e s ta t i q u e , re la t i f s à l ' a rc h i t e c tu re . A ca d em ie R o ya le d es Sc ien ce s , P a r is .

M e m o ire s d e M a th e m a t iq u e e t d e P h y s iq u e . 7 : 3 4 3 -3 8 2 .

C o x , A . a n d H a r t , R . B . 1 9 8 6 . P la te T ec to n ic s : H o w i t W o r k s . B l a c k w e l l S c i e n t i f i c : P a l o

A l to , C A . 3 9 2 p a g e s .

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d e fo rm e d g ra n o d io r i t e d y k e s n o r t h o f H o ls te in s b o rg , W e s t G re e n la n d . J o u rn a l o f th e

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N Y . P a g e s 2 8 1 -2 9 9 .

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p a l e o s t r e s s f i e l d s d u r in g th e d e v e l o p m e n t o f th e A p p a l a c h i a n P l a te a u , N e w Y o rk .

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c o n te m p o ra ry s t r e s s w i th in th e l i t h o s p h e re o f N o r th A m e r i c a ? T e c to n ic s . 1 : 1 6 1 -1 7 7 .

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s t r e s s t e n s o r u s in g e a r th q u a k e fo c a l m e c h a n i sm d a t a : A p p l i c a t i o n to th e S a n F e r n a n d o

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la m e l l a e in a m in o r fo ld . A m e r ic a n J o u rn a l o f S c ie n c e . 2 6 3 : 7 2 9 -7 4 6 .

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S p a n g , J . H . 1 9 7 2 . N u m e r i c a l m e th o d fo r d y n a m ic a n a ly s i s o f c a l c i t e tw in la m e l l a e .

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S p a n g , J . H . a n d V a n D e r L e e , J . 1 9 7 5 . N u m e r i c a l d y n a m i c a n a ly s i s o f q u a r t z d e fo rm a t io n

la m e l l a e a n d c a lc i t e a n d d o lo m i t e tw in la m e l l a e . G e o lo g ic a l S o c ie ty o f A m e r ic a

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T j i a , H . D . 1 9 7 2 . F a u l t m o v e m e n t , r e o r i e n te d s t r e s s f i e ld a n d s u b s id ia ry s t r u c tu re s . P a c i f ic

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B r i t i sh C o lu m b ia : S t r ik e s l ip s t ru c tu re s o v e r p r in t in g m id - C re t a c e o u s t h ru s t s t ru c tu r e s .

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5 9 :1 1 8 -1 3 0 .

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o r ie n t a t io n o f a p a l e o - s t r e s s e l l i p s o i d f ro m th e d i s t r i b u t i o n o f s l i c k e n s id e s t r i a t io n s .

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M in é ra lo g ie . 1 0 2 : 2 1 0 -2 1 5 .

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8 4 :3 5 0 -3 5 5 .

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J o u r n a l o f G e o p h y s ic a l R e se a rc h . 8 5 : 6 1 1 3 -6 1 5 6 .

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